Chapter 3 Linear Mappings and Matrices: Exercises

import numpy as np
np.set_printoptions(precision=4, suppress=True, linewidth=100)

import matplotlib.pyplot as plt
# Fonts: the English edition needs no CJK font, so nothing is downloaded when _lang == 'en';
# the Chinese editions use this same block to fetch Noto Sans TC/SC where no CJK font is installed
import os, urllib.request
import matplotlib.font_manager as fm
_lang = 'en'
_cjk = ['Microsoft JhengHei', 'PingFang TC', 'Noto Sans CJK TC', 'Noto Sans TC']
_have = {f.name for f in fm.fontManager.ttflist}
if _lang != 'en' and not _have & set(_cjk):
    _font = os.path.join(os.path.expanduser('~'), '.cache', 'fonts', 'NotoSansTC.ttf')
    try:
        if not os.path.exists(_font):
            os.makedirs(os.path.dirname(_font), exist_ok=True)
            urllib.request.urlretrieve('https://github.com/google/fonts/raw/main/ofl/notosanstc/NotoSansTC%5Bwght%5D.ttf', _font + '.part')
            os.replace(_font + '.part', _font)
        fm.fontManager.addfont(_font)
        _have.add('Noto Sans TC')
    except OSError as err:
        print('Could not download the CJK font; Chinese text in figures may not display:', err)
plt.rcParams['font.family'] = [f for f in _cjk if f in _have] + ['DejaVu Sans']
plt.rcParams['axes.unicode_minus'] = False

# ── Visual hierarchy helpers ─────────────────────────────────────────────
def print_header(title):
    print("=" * 60)
    print(f"  {title}")
    print("=" * 60)

def print_step(step, desc):
    print(f"\n▶ Step {step}: {desc}")
    print("-" * 40)

def compare_print(label, actual, expected_desc):
    print(f"\n[{label}]")
    print(f"  Computed value: {actual}")
    print(f"  Expected value: {expected_desc}")

# ── Math utilities ─────────────────────────────────────────────
def proj_vec(u, v):
    """Projection vector of v onto the direction of u"""
    u = np.asarray(u, float); v = np.asarray(v, float)
    return (np.dot(u, v) / np.dot(u, u)) * u

def proj_mat(u):
    """Projection matrix P = uu^T / (u^T u)"""
    u = np.asarray(u, float).reshape(-1, 1)
    return u @ u.T / (u.T @ u)[0, 0]

def reflect_mat_2d(n):
    """Reflection matrix I − 2nn^T in ℝ², where n is a unit normal vector"""
    n = np.asarray(n, float); n = n / np.linalg.norm(n)
    return np.eye(2) - 2 * np.outer(n, n)

def reflect_mat_3d(n):
    """Reflection matrix I − 2nn^T in ℝ³"""
    n = np.asarray(n, float); n = n / np.linalg.norm(n)
    return np.eye(3) - 2 * np.outer(n, n)

def rodrigues(u, theta):
    """Rodrigues rotation matrix"""
    u = np.asarray(u, float); u = u / np.linalg.norm(u)
    K = np.array([[0, -u[2], u[1]], [u[2], 0, -u[0]], [-u[1], u[0], 0]])
    return np.eye(3) + np.sin(theta)*K + (1 - np.cos(theta))*(K @ K)

# ── Complex number → matrix mapping ──────────────────────────
def M(z):
    """The 2×2 matrix M(z) = [[a,-b],[b,a]] corresponding to the complex number z = a + bi"""
    a, b = z.real, z.imag
    return np.array([[a, -b], [b, a]])

# ── Skew-symmetric matrix of a vector ─────────────────────
def skew(v):
    """The skew-symmetric matrix [v]× corresponding to the 3D vector v = (a,b,c)"""
    a, b, c = v
    return np.array([[0, -c, b],
                     [c,  0, -a],
                     [-b, a,  0]])
print("Helper functions loaded")

Matrix Multiplication and Matrix Powers¶

Exercise 1 (A Product with the Zero Matrix)

Compute $(\mathbf{A}\mathbf{B})\mathbf{C}$ , where

\mathbf{A} = \begin{pmatrix}12 & -8 \\ 7 & -24\end{pmatrix},\quad \mathbf{B} = \begin{pmatrix}11 & 13 \\ -21 & -30\end{pmatrix},\quad \mathbf{C} = \begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix}

(1)

Hint. Note that $\mathbf{C}$ is the zero matrix; think first about the effect of $\mathbf{C}$ .

Exercise 2 (Distributivity:

AB+AC

)

Find $\mathbf{A}\mathbf{B} + \mathbf{A}\mathbf{C}$ , where

\mathbf{A} = \begin{pmatrix}33 & -18 \\ 27 & -243\end{pmatrix},\quad \mathbf{B} = \begin{pmatrix}4 & 7 \\ -6 & 12\end{pmatrix},\quad \mathbf{C} = \begin{pmatrix}-3 & -7 \\ 6 & -11\end{pmatrix}

(3)

Hint. Use distributivity, $\mathbf{A}(\mathbf{B}+\mathbf{C})=\mathbf{A}\mathbf{B}+\mathbf{A}\mathbf{C}$ , to reduce the amount of computation.

Exercise 3 (Products of Rectangular Matrices)

Compute the products $\mathbf{A}\mathbf{B}$ and $\mathbf{B}\mathbf{A}$ (if they are defined), where

\mathbf{A} = \begin{pmatrix}2 & -1 & 3 \\ 0 & 4 & -2\end{pmatrix},\quad \mathbf{B} = \begin{pmatrix}1 & 0 \\ -1 & 2 \\ 3 & 1\end{pmatrix}

(6)

Hint. $\mathbf{A}$ is $2\times 3$ and $\mathbf{B}$ is $3\times 2$ ; first check for which products the dimensions match.

Exercise 4 (

AB=BA

?)

Compute $\mathbf{A}\mathbf{B}$ and $\mathbf{B}\mathbf{A}$ . Is $\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}$ ?

\mathbf{A} = \begin{pmatrix}1&0&2\\-1&3&1\\2&1&0\end{pmatrix},\quad \mathbf{B} = \begin{pmatrix}2&1&0\\0&-1&3\\1&2&1\end{pmatrix}

(10)

Hint. In general, multiplication of square matrices is not commutative.

Exercise 5 (Multiplication by the Identity Matrix)

Let $\mathbf{I}_3$ be the identity matrix of order 3. Compute $\mathbf{I}_3\mathbf{A}$ and $\mathbf{A}\mathbf{I}_3$ . Is $\mathbf{I}_3\mathbf{A}=\mathbf{A}\mathbf{I}_3$ ?

\mathbf{A} = \begin{pmatrix}2&-1&3\\1&0&2\\-1&4&1\end{pmatrix}

(14)

Hint. The identity matrix is the multiplicative identity for matrix multiplication.

Exercise 6 (A Diagonal Matrix and a Constant Matrix)

Compute $\mathbf{A}\mathbf{B}$ and $\mathbf{B}\mathbf{A}$ . Is $\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}$ ?

\mathbf{A} = \begin{pmatrix}-2&0&0&0\\0&0&0&0\\0&0&2&0\\0&0&0&4\end{pmatrix},\quad \mathbf{B} = \begin{pmatrix}1&1&1&1\\1&1&1&1\\1&1&1&1\\1&1&1&1\end{pmatrix}

(17)

Hint. Products of a diagonal matrix and a constant matrix follow a simple pattern.

Exercise 7 (Products of Constant Matrices of Order

n

)

Let $\mathbf{A}$ , $\mathbf{B}$ be square matrices of order $n$ all of whose entries are the constants $a$ and $b$ , respectively. Find $\mathbf{A}\mathbf{B}$ .

\mathbf{A} = \begin{pmatrix}a&a&\cdots&a\\a&a&\cdots&a\\\vdots&\vdots&\ddots&\vdots\\a&a&\cdots&a\end{pmatrix},\quad \mathbf{B} = \begin{pmatrix}b&b&\cdots&b\\b&b&\cdots&b\\\vdots&\vdots&\ddots&\vdots\\b&b&\cdots&b\end{pmatrix}

(21)

Hint. Look at the pattern in the computation of $(\mathbf{A}\mathbf{B})_{ij}$ ; each entry is a sum of $n$ terms.

# ============================================================
print_header("Matrix Multiplication")
# ============================================================

print_step(1, "Exercise 1: (AB)C = 0 (AB times the zero matrix)")
A1 = np.array([[12,-8],[7,-24]]); B1 = np.array([[11,13],[-21,-30]]); C1 = np.zeros((2,2))
result = (A1 @ B1) @ C1
compare_print("(AB)C", result, "zero matrix O")

print_step(2, "Exercise 2: check distributivity AB+AC = A(B+C)")
A2 = np.array([[33,-18],[27,-243]])
B2 = np.array([[4,7],[-6,12]]); C2 = np.array([[-3,-7],[6,-11]])
lhs = A2@B2 + A2@C2; rhs = A2@(B2+C2)
print("  AB+AC ="); print(lhs)
compare_print("AB+AC = A(B+C)?", np.allclose(lhs, rhs), "True")

print_step(3, "Exercise 3: the products AB and BA of rectangular matrices")
A3 = np.array([[2,-1,3],[0,4,-2]]); B3 = np.array([[1,0],[-1,2],[3,1]])
print("  AB ="); print(A3@B3)
print("  BA ="); print(B3@A3)

print_step(4, "Exercise 4: AB and BA (a counterexample to commutativity)")
A4 = np.array([[1,0,2],[-1,3,1],[2,1,0]]); B4 = np.array([[2,1,0],[0,-1,3],[1,2,1]])
print("  AB ="); print(A4@B4)
print("  BA ="); print(B4@A4)
compare_print("AB = BA?", np.allclose(A4@B4, B4@A4), "False (square matrices do not commute in general)")

print_step(5, "Exercise 5: I₃A = A (multiplication by the identity matrix)")
A5 = np.array([[2,-1,3],[1,0,2],[-1,4,1]]); I3 = np.eye(3)
compare_print("I₃A = A?", np.allclose(I3@A5, A5), "True")

print_step(6, "Exercise 6: diagonal matrix × all-ones matrix (AB and BA)")
A6 = np.diag([-2,0,2,4]); B6 = np.ones((4,4))
print("  AB ="); print(A6@B6)
print("  BA ="); print(B6@A6)

print_step(7, "Exercise 7: constant matrices of order n, aJ·bJ = nabJ")
n=4; a,b=3,2
An=a*np.ones((n,n)); Bn=b*np.ones((n,n))
compare_print(f"AB = nab·J (n={n},a={a},b={b})",
    np.allclose(An@Bn, n*a*b*np.ones((n,n))), "True")

Exercise 8 (A Matrix Polynomial)

Compute $\mathbf{A}^2 - 7\mathbf{A} + 12\mathbf{I}_2$ , where

\mathbf{A} = \begin{pmatrix}-2&-3\\10&9\end{pmatrix}

(24)

Hint. First compute $\mathbf{A}^2$ , then substitute into the polynomial.

Exercise 9 (Powers of a Rotation Matrix)

Compute $\mathbf{A}^2$ , $\mathbf{A}^3$ , $\mathbf{A}^4$ , where

\mathbf{A} = \begin{pmatrix}0&1\\-1&0\end{pmatrix}

(27)

Hint. $\mathbf{A}$ is a rotation matrix; look for the periodic pattern in its powers.

Exercise 10 (A Nilpotent Matrix)

Compute $\mathbf{A}^2$ , $\mathbf{A}^3$ , and $\mathbf{A}^k$ ( $k\geq 1$ ), where

\mathbf{A} = \begin{pmatrix}1&1&1\\0&0&0\\-1&-1&-1\end{pmatrix}

(29)

Hint. After computing $\mathbf{A}^2$ , observe the linear relations among the rows.

Exercise 11 (Powers of a Diagonal Matrix)

Let $\mathbf{A} = \mathrm{diag}(\lambda_0,\lambda_1,\ldots,\lambda_{n-1})$ be a diagonal matrix of order $n$ . Find $\mathbf{A}^k$ ( $k\geq 1$ ).

Hint. A product of diagonal matrices is again a diagonal matrix, with the diagonal entries multiplied one by one.

# ============================================================
print_header("Matrix Powers")
# ============================================================

print_step(1, "Exercise 8: the matrix polynomial A²−7A+12I = 0")
A = np.array([[-2,-3],[10,9]])
result = A@A - 7*A + 12*np.eye(2)
compare_print("A²−7A+12I", result, "zero matrix O")

print_step(2, "Exercise 9: powers of a rotation matrix (A = clockwise 90°)")
A = np.array([[0,1],[-1,0]])
print("  A² ="); print(A@A)
print("  A³ ="); print(A@A@A)
print("  A⁴ ="); print(np.linalg.matrix_power(A,4))
compare_print("A⁴", np.linalg.matrix_power(A,4), "I (rotation by 360° = identity)")

print_step(3, "Exercise 10: the nilpotent matrix A²=0")
A = np.array([[1,1,1],[0,0,0],[-1,-1,-1]])
compare_print("A²", A@A, "zero matrix O (index of nilpotency = 2)")

print_step(4, "Exercise 11: powers of a diagonal matrix Aᵏ = diag(λ₀ᵏ,…)")
lam = np.array([2.0, -1.0, 3.0]); A = np.diag(lam); k=5
print(f"  A^{k} ="); print(np.linalg.matrix_power(A,k))
compare_print(f"diag(λᵢ^{k})", np.diag(lam**k), "same as A^k")

Exercise 12 (◆The Cyclic Property of the Trace)

Prove that the trace of a matrix has the following properties:

Let $\mathbf{A}$ be an $m\times n$ matrix and $\mathbf{B}$ an $n\times m$ matrix. Prove that

\text{tr}(\mathbf{AB}) = \text{tr}(\mathbf{BA})

(33)

Note that equation (33) holds even when $\mathbf{AB} \neq \mathbf{BA}$ (matrix multiplication is not commutative).

Let $\mathbf{A}$ , $\mathbf{B}$ , $\mathbf{C}$ have sizes $m\times n$ , $n\times p$ , $p\times m$ , respectively, so that all three cyclic products are square matrices. Prove that

\text{tr}(\mathbf{ABC}) = \text{tr}(\mathbf{BCA}) = \text{tr}(\mathbf{CAB})

(34)

# ============================================================
print_header("The Cyclic Property of the Trace")
# ============================================================

np.random.seed(42)

print_step(1, "Verify tr(AB) = tr(BA) (3×4 and 4×3 matrices)")
A = np.random.randn(3, 4)
B = np.random.randn(4, 3)
trAB = np.trace(A @ B)
trBA = np.trace(B @ A)
compare_print("tr(AB)", f"{trAB:.6f}", f"tr(BA) = {trBA:.6f}")
print(f"  equal? {np.isclose(trAB, trBA)}")

print_step(2, "Verify the cyclic property for three matrices")
A = np.random.randn(3, 3)
B = np.random.randn(3, 3)
C = np.random.randn(3, 3)
trABC = np.trace(A @ B @ C)
trBCA = np.trace(B @ C @ A)
trCAB = np.trace(C @ A @ B)
trACB = np.trace(A @ C @ B)   # not a cyclic permutation (usually unequal)
print(f"  tr(ABC) = {trABC:.6f}")
print(f"  tr(BCA) = {trBCA:.6f}")
print(f"  tr(CAB) = {trCAB:.6f}")
print(f"  tr(ACB) = {trACB:.6f}  ← not a cyclic permutation (unequal in general)")
print(f"\n  tr(ABC)=tr(BCA)? {np.isclose(trABC, trBCA)}")
print(f"  tr(ABC)=tr(CAB)? {np.isclose(trABC, trCAB)}")
print(f"  tr(ABC)=tr(ACB)? {np.isclose(trABC, trACB)} ← a nonzero difference is expected")

print_step(3, "AB ≠ BA, but tr(AB) = tr(BA)")
# verify that for square matrices AB≠BA but the traces are equal
A2 = np.array([[1.,2.],[3.,4.]])
B2 = np.array([[0.,1.],[1.,0.]])
print(f"  A·B =\n{A2@B2}")
print(f"  B·A =\n{B2@A2}")
print(f"  A·B ≠ B·A? {not np.allclose(A2@B2, B2@A2)}")
print(f"  tr(AB) = {np.trace(A2@B2):.4f} = tr(BA) = {np.trace(B2@A2):.4f}? {np.isclose(np.trace(A2@B2), np.trace(B2@A2))}")

Linear Mappings and Matrix Representations¶

Exercise 13 (Images of the Basis Vectors under

T_A

)

Let $\mathbf{A} = \begin{pmatrix}2&-1\\-3&4\end{pmatrix}$ , with corresponding linear mapping $T_A:\mathbb{R}^2\to\mathbb{R}^2$ .

Find $T_A(\mathbf{e}_0)$ and $T_A(\mathbf{e}_1)$ , where $\mathbf{e}_0=(1,0)$ , $\mathbf{e}_1=(0,1)$ .
Find $T_A(3,-2)$ and express it as a linear combination of $T_A(\mathbf{e}_0)$ , $T_A(\mathbf{e}_1)$ .

Hint. $T_A(\mathbf{e}_i)$ is exactly column $i$ of the matrix $\mathbf{A}$ .

Exercise 14 (

T_A:\mathbb{R}^3\to\mathbb{R}^2

)

Let $\mathbf{A} = \begin{pmatrix}2&1&0\\0&3&-1\end{pmatrix}$ , with corresponding linear mapping $T_A:\mathbb{R}^3\to\mathbb{R}^2$ .

Find $T_A(\mathbf{e}_0)$ , $T_A(\mathbf{e}_1)$ , $T_A(\mathbf{e}_2)$ .
Use a linear combination to find $T_A(-1,2,6)$ .

Hint. The output of a linear mapping is a linear combination of the images of the basis vectors.

Exercise 15 (Finding a Preimage)

Let $\mathbf{A} = \begin{pmatrix}1&1&1\\0&1&1\\0&0&1\end{pmatrix}$ , with corresponding linear mapping $T_A:\mathbb{R}^3\to\mathbb{R}^3$ .

Can we find a vector $\mathbf{u}$ in $\mathbb{R}^3$ such that $T_A(\mathbf{u})=(5,9,4)$ ?

Hint. Solve the system of linear equations $\mathbf{A}\mathbf{u}=\mathbf{b}$ ; is $\mathbf{A}$ invertible?

Exercise 16 (Vectors in the Kernel)

Let $\mathbf{A} = \begin{pmatrix}1&2&3\\2&4&6\end{pmatrix}$ , with corresponding linear mapping $T_A:\mathbb{R}^3\to\mathbb{R}^2$ .

Observe the relations among the column vectors of $\mathbf{A}$ , and find a nonzero vector $\mathbf{u}\in\mathbb{R}^3$ such that $T_A(\mathbf{u})=\mathbf{0}$ .

Hint. Note that row 1 of $\mathbf{A}$ is a multiple of row 0.

Exercise 17 (The Matrix from the Images of the Basis Vectors,

\mathbb{R}^2\to\mathbb{R}^3

)

A linear mapping $T:\mathbb{R}^2\to\mathbb{R}^3$ satisfies

T(1,0)=(3,-1,7),\quad T(0,1)=(2,5,-2)

(43)

Find the matrix representation $\mathbf{A}$ of $T$ , and compute $T(4,-3)$ .

Hint. Column $i$ of the matrix is exactly $T(\mathbf{e}_i)$ .

Exercise 18 (The Matrix from the Images of the Basis Vectors,

\mathbb{R}^2\to\mathbb{R}^4

)

A linear mapping $T:\mathbb{R}^2\to\mathbb{R}^4$ satisfies

T(1,0)=(3,-1,7,5),\quad T(1,1)=(2,5,-2,6)

(46)

Find the matrix representation $\mathbf{A}$ of $T$ , and compute $T(-1,1)$ .

Hint. First use $T(1,0)$ and $T(1,1)$ to obtain $T(0,1)=T(1,1)-T(1,0)$ .

Exercise 19 (Linearity of a Projection Mapping)

Consider the mapping $T:\mathbb{R}^3\to\mathbb{R}^2$ , $T(a,b,c)=(a,b)$ (projection onto the first two components).

Explain why $T$ satisfies the linearity conditions.
Find $T(\mathbf{e}_0)$ , $T(\mathbf{e}_1)$ , $T(\mathbf{e}_2)$ , and write down the matrix representation $\mathbf{A}$ .
Decide whether the mapping $T^2=T\circ T$ and the matrix $\mathbf{A}^2$ are defined, and explain why.

Hint. $T^2 = T\circ T$ ; check whether the output of the first mapping can serve as the input of the second, and whether the inner dimensions of the matrix product agree.

Exercise 20 (Reflection across the

xy

-Plane)

Consider the mapping $T:\mathbb{R}^3\to\mathbb{R}^3$ , $T(a,b,c)=(a,b,-c)$ (reflection across the $xy$ -plane).

Explain why $T$ is a linear mapping, and find its matrix representation $\mathbf{A}$ .
Compute the matrix of $T^2$ and compare it with $\mathbf{A}^2$ .

Hint. Reflecting twice should bring every point back to where it started.

Exercise 21 (Rotation by

90°

about the

z

-Axis)

The linear mapping $T:\mathbb{R}^3\to\mathbb{R}^3$ rotates points of space counterclockwise by $90°$ ( $\frac{\pi}{2}$ ) about the $z$ -axis.

Find $T(\mathbf{e}_0)$ , $T(\mathbf{e}_1)$ , $T(\mathbf{e}_2)$ , and write down the matrix $\mathbf{A}$ .
Find the matrix of $T^4$ and compare it with $\mathbf{A}^4$ .

Hint. Rotating $\mathbf{e}_0=(1,0,0)$ by $90°$ about the $z$ axis gives $(0,1,0)$ .

Exercise 22 (The Shift Operator)

A linear mapping $T:\mathbb{R}^4\to\mathbb{R}^4$ satisfies

T(\mathbf{e}_0)=\mathbf{e}_1,\quad T(\mathbf{e}_1)=\mathbf{e}_2,\quad T(\mathbf{e}_2)=\mathbf{e}_3,\quad T(\mathbf{e}_3)=\mathbf{0}

(54)

Write down the matrix $\mathbf{A}$ .
Compute the matrices of $T^2$ , $T^3$ , $T^4$ ; for $k\geq 4$ , describe $T^k$ .

Hint. $T$ is called a shift operator; it is nilpotent.

# ============================================================
print_header("Linear Mappings and Matrix Representations")
# ============================================================

print_step(1, "Exercise 13(2): T_A(3,−2), A ∈ ℝ²×²")
A = np.array([[2,-1],[-3,4]])
result = A @ np.array([3,-2])
print("  T_A(3,−2) =", result)

print_step(2, "Exercise 14(2): T_A(−1,2,6), A ∈ ℝ²×³")
A = np.array([[2,1,0],[0,3,-1]])
result = A @ np.array([-1,2,6])
print("  T_A(−1,2,6) =", result)

print_step(3, "Exercise 15: finding a preimage")
A = np.array([[1,1,1],[0,1,1],[0,0,1]])
b = np.array([5,9,4])
u = np.linalg.solve(A, b)
print("  preimage u =", u)
compare_print("check Au", A@u, b)

print_step(4, "Exercise 16: kernel check")
A = np.array([[1,2,3],[2,4,6]])
u = np.array([3,0,-1])
compare_print("Au (should be 0)", A@u, "[0, 0]")

print_step(5, "Exercise 17: building the matrix from the images of the basis vectors, ℝ²→ℝ³")
A = np.array([[3,2],[-1,5],[7,-2]])
result = A @ np.array([4,-3])
print("  T(4,−3) =", result)

print_step(6, "Exercise 18: building the matrix from the images of the basis vectors, ℝ²→ℝ⁴")
Te0 = np.array([3,-1,7,5]); Te01 = np.array([2,5,-2,6])
Te1 = Te01 - Te0
A = np.column_stack([Te0, Te1])
result = A @ np.array([-1,1])
print("  T(−1,1) =", result)

print_step(7, "Exercise 21(2): powers of the rotation by 90° about the z axis")
A_rot = np.array([[0,-1,0],[1,0,0],[0,0,1]])
compare_print("A_rot⁴", np.linalg.matrix_power(A_rot, 4), "I (rotation by 4×90°=360°)")

print_step(8, "Exercise 22(2): the shift operator A_shift⁴")
A_sh = np.array([[0,0,0,0],[1,0,0,0],[0,1,0,0],[0,0,1,0]])
compare_print("A_shift⁴", np.linalg.matrix_power(A_sh,4), "zero matrix (nilpotent)")

Scaling Mappings¶

Exercise 23 (Scaling Mappings:

\mathbb{R}^2

and

\mathbb{R}^3

)

(a) In $\mathbb{R}^2$ , the scaling mapping $T$ stretches the $x$ direction by a factor of 2 and the $y$ direction by a factor of 3.

Write down the matrix $\mathbf{A}$ .
Find $T(-3,4)$ .
A rectangle of area 2 is mapped by $T$ ; what is the area of its image?
Find $\mathbf{A}^{-1}$ . Is the corresponding mapping also a scaling mapping?

(b) In $\mathbb{R}^3$ , the $x$ direction is shrunk by a factor of $\frac{1}{10}$ , $y$ is stretched by a factor of 2, and $z$ by a factor of 4.

Write down the matrix $\mathbf{A}$ .
Find $T(-2,1,3)$ .
A cuboid of volume 20 is mapped by $T$ ; what is the volume of its image?
Find $\mathbf{A}^{-1}$ . Is the corresponding mapping also a scaling mapping?

Hint. The matrix of a scaling mapping is diagonal, and its determinant gives the area/volume scale factor.

Exercise 24 (The Image of the Unit Circle)

In $\mathbb{R}^2$ , the scaling mapping $T$ has matrix $\mathbf{A}=\begin{pmatrix}2&0\\0&1\end{pmatrix}$ .

What figure does the unit circle $x^2+y^2=1$ become under $T$ ? Write down its equation.

Hint. Let $(x',y')=T(x,y)$ and substitute $x=x'/2$ , $y=y'$ into the original equation.

Exercise 25 (Composition of Scaling Mappings)

Two scaling mappings in $\mathbb{R}^2$ :

S_0:\text{matrix }\begin{pmatrix}2&0\\0&3\end{pmatrix},\quad S_1:\text{matrix }\begin{pmatrix}4&0\\0&1/2\end{pmatrix}

(60)

Compute the matrix of the composite mapping $S_1\circ S_0$ (first apply $S_0$ , then $S_1$ ).
Explain why $S_1\circ S_0$ is also a scaling mapping, and how its scale factors are related to the original ones.
If $S_1=S_0^{-1}$ , what is the matrix of $S_1$ ?

Hint. The matrix of the composite mapping $=S_1$ ’s matrix $\times S_0$ ’s matrix.

Exercise 26 (An Area-Preserving Scaling Mapping)

In $\mathbb{R}^2$ , the matrix of a scaling mapping is $\mathbf{A}=\begin{pmatrix}1000&0\\0&x\end{pmatrix}$ .

If every figure has the same area after being mapped by $T$ as before, what must $x$ be?

Hint. The area scale factor is $=|\det(\mathbf{A})|=1$ .

# ============================================================
print_header("Scaling Mappings")
# ============================================================

print_step(1, "Exercise 23(a): scaling diag(2,3) in ℝ²")
A = np.diag([2.0, 3.0])
print("  T(−3,4) =", A @ np.array([-3,4]))
compare_print("area scale factor |det(A)|", abs(np.linalg.det(A)), "6 (area of the unit square × 6)")
print("  A⁻¹ ="); print(np.linalg.inv(A))

print_step(2, "Exercise 23(b): scaling diag(0.1,2,4) in ℝ³")
A3 = np.diag([0.1, 2.0, 4.0])
print("  T(−2,1,3) =", A3 @ np.array([-2,1,3]))
compare_print("volume scale factor |det(A)|", abs(np.linalg.det(A3)), "0.8")

print_step(3, "Exercise 25(1): composition of scaling mappings S₁∘S₀")
A1 = np.diag([2.0, 3.0]); A2 = np.diag([4.0, 0.5])
composed = np.diag(A2 @ A1)
compare_print("scale factors of S₁∘S₀", composed, "diag(8, 1.5)")

print_step(4, "Exercise 26: area-preserving scaling, det = 1")
compare_print("det(diag(1000, 0.001))", np.linalg.det(np.diag([1000, 1/1000])), "1")

Projection Mappings¶

Exercise 27 (Projection onto a Line in

\mathbb{R}^2

)

In $\mathbb{R}^2$ , consider the mapping $T$ that projects onto the line spanned by the vector $\mathbf{v}=(3,4)$ .

Find the unit vector $\mathbf{u}$ of $\mathbf{v}$ .
Use the formula $\mathbf{P}=\mathbf{u}\mathbf{u}^\top$ to find the projection matrix.
Compute the projection $T(5,0)$ of $(5,0)$ onto $\mathbf{u}$ .
Verify that $T(5,0)$ and $(5,0)-T(5,0)$ are perpendicular to each other.
Verify that $\mathbf{P}^2=\mathbf{P}$ and $\mathbf{P}^\top=\mathbf{P}$ .

Hint. To compute $\mathbf{u}\mathbf{u}^\top$ , $\mathbf{u}$ must be written as a column vector (a $2\times1$ matrix).

Exercise 28 (Projection onto a Line in

\mathbb{R}^3

)

In $\mathbb{R}^3$ , let $T$ be the mapping that projects onto the line spanned by the vector $\mathbf{v}=(1,-2,-2)$ .

Find the projection matrix $\mathbf{P}$ .
Verify that $\mathbf{P}^2=\mathbf{P}$ and $\mathbf{P}^\top=\mathbf{P}$ .
First think about what $\det(\mathbf{P})$ should be, then compute it to check.

Hint. For a projection onto a one-dimensional subspace, $\operatorname{rank}(\mathbf{P})=1$ , so $\det(\mathbf{P})=0$ .

Exercise 29 (Projection onto a Line in

\mathbb{R}^4

)

In $\mathbb{R}^4$ , let $T$ be the mapping that projects onto the line spanned by the vector $\mathbf{v}=(1,1,1,1)$ .

Find the projection matrix $\mathbf{P}$ .
Verify that $T(\mathbf{u})\cdot(\mathbf{u}-T(\mathbf{u}))=0$ (let $\mathbf{u}=(a,b,c,d)$ ).
Verify that $\mathbf{P}^2=\mathbf{P}$ and $\mathbf{P}^\top=\mathbf{P}$ .

Hint. The vector $\mathbf{v}=(1,1,1,1)$ has length 2.

Exercise 30 (Projection onto the

xy

-Plane)

In $\mathbb{R}^3$ , let $T$ be the linear mapping that projects onto the $xy$ -plane ( $z=0$ ).

Write down the matrix $\mathbf{P}$ directly from the projections of the $\mathbf{e}_i$ .
Verify that $T(a,b,c)$ is perpendicular to $(a,b,c)-T(a,b,c)$ .
Verify that $\mathbf{P}^2=\mathbf{P}$ and $\mathbf{P}^\top=\mathbf{P}$ .
Compute $\det(\mathbf{P})$ and explain what it means.

Hint. The normal vector of the $xy$ -plane is $\mathbf{e}_2=(0,0,1)$ .

Exercise 31 (Projection onto the Plane

x+y+z=0

)

In $\mathbb{R}^3$ , let $T$ be the mapping that projects onto the plane $x+y+z=0$ .

Find the unit normal vector $\mathbf{n}$ of the plane.
Use $\mathbf{P}=\mathbf{I}_3-\mathbf{n}\mathbf{n}^\top$ to find the projection matrix.
Verify that $\mathbf{P}^2=\mathbf{P}$ and $\mathbf{P}^\top=\mathbf{P}$ .
Compute $T(1,0,0)$ and verify that the result lies in the plane, and that $(0,1,-1)-T(0,1,-1)$ is parallel to the normal vector.

Hint. Projection onto the plane $=$ the identity mapping $-$ projection onto the normal vector.

Exercise 32 (Distance from a Point to a Line)

In $\mathbb{R}^2$ , let $L$ be the line through the origin with direction vector $\mathbf{v}=(1,2)$ .

Use the idea of a projection mapping to find the distance from the point $A(6,8)$ to $L$ .

Hint. The distance is $=\|\mathbf{a}-T(\mathbf{a})\|$ , where $T(\mathbf{a})$ is the projection of $\mathbf{a}$ onto $L$ .

Exercise 33 (The Distance between Parallel Lines)

In $\mathbb{R}^3$ , $L_0$ passes through the origin and $L_1$ passes through $(2,2,1)$ , and both have direction vector $\mathbf{v}=(1,0,3)$ .

Use a projection mapping to find the distance between $L_0$ and $L_1$ .

Hint. Take the point $\mathbf{p}=(2,2,1)$ on $L_1$ ; the distance is $=\|\mathbf{p}-T(\mathbf{p})\|$ ( $T$ is the projection onto $L_0$ ).

Exercise 34 (Composition of Projection Matrices)

In $\mathbb{R}^3$ , $\mathbf{P}_0$ projects onto the $xy$ -plane and $\mathbf{P}_1$ projects onto the $xz$ -plane.

Write down $\mathbf{P}_0$ and $\mathbf{P}_1$ .
Compute $\mathbf{P}_1\mathbf{P}_0$ (first apply $\mathbf{P}_0$ , then $\mathbf{P}_1$ ).
What is the geometric meaning of $\mathbf{P}_1\mathbf{P}_0$ ? Is it a projection mapping?
Compare $\mathbf{P}_1\mathbf{P}_0$ and $\mathbf{P}_0\mathbf{P}_1$ . Does the order matter?

Hint. Does $\mathbf{P}_1\mathbf{P}_0$ satisfy $(\mathbf{P}_1\mathbf{P}_0)^2=\mathbf{P}_1\mathbf{P}_0$ ?

# ============================================================
print_header("Projection Mappings")
# ============================================================

print_step(1, "Exercise 27: projection onto the direction (3,4), checking idempotence")
v = np.array([3.0, 4.0]); u = v / np.linalg.norm(v)
P = np.outer(u, u)
a = np.array([5.0, 0])
Ta = P @ a
print("  T(5,0) =", Ta)
compare_print("(a−Ta)⊥Ta", f"{np.dot(Ta, a-Ta):.2e}", "0")
compare_print("P² = P (idempotence)", np.allclose(P@P, P), "True")

print_step(2, "Exercise 28: projection onto the direction (1,−2,−2) (ℝ³)")
v = np.array([1.0, -2, -2]); u = v / np.linalg.norm(v)
P = np.outer(u, u)
print("  P ="); print(P)
compare_print("det(P) (should be 0, rank 1)", f"{np.linalg.det(P):.2e}", "0")
compare_print("P² = P", np.allclose(P@P, P), "True")

print_step(3, "Exercise 30(4): projection onto the xy-plane")
P_xy = np.diag([1.0, 1, 0])
compare_print("det(P_xy) (rank 2)", np.linalg.det(P_xy), "0")

print_step(4, "Exercise 31: projection onto the plane x+y+z=0 (perpendicular to the normal vector n=(1,1,1)/√3)")
n = np.array([1.0, 1, 1]) / np.sqrt(3)
P_plane = np.eye(3) - np.outer(n, n)
print("  P_plane ="); print(np.round(P_plane, 4))
compare_print("P² = P (idempotent)", np.allclose(P_plane@P_plane, P_plane), "True")
print("  T(1,0,0) =", P_plane @ np.array([1,0,0]))

print_step(5, "Exercise 32: distance from the point (6,8) to the line span{(1,2)}")
v = np.array([1.0, 2]); u = v / np.linalg.norm(v)
P = np.outer(u, u)
a = np.array([6.0, 8])
Ta = P @ a
dist = np.linalg.norm(a - Ta)
compare_print("distance ‖a−P(a)‖", f"{dist:.6f}", f"4/√5 = {4/np.sqrt(5):.6f}")

print_step(6, "Exercise 33: distance between parallel lines")
v = np.array([1.0, 0, 3]); u = v / np.linalg.norm(v)
P = np.outer(u, u)
p = np.array([2.0, 2, 1])
dist = np.linalg.norm(p - P @ p)
compare_print("distance", f"{dist:.6f}", f"√26/2 = {np.sqrt(26)/2:.6f}")

print_step(7, "Exercise 34: composition of two projection matrices (P₀P₁ = P₁P₀)")
P0 = np.diag([1.0,1,0]); P1 = np.diag([1.0,0,1])
print("  P₁P₀ ="); print(P1@P0)
print("  P₀P₁ ="); print(P0@P1)
compare_print("P₀P₁ = P₁P₀?", np.allclose(P0@P1, P1@P0), "True (diagonal matrices commute)")

Reflection Mappings¶

Exercise 35 (Reflection across a Line in

\mathbb{R}^2

)

In $\mathbb{R}^2$ , let $L$ be the line through the origin with normal vector $\mathbf{N}=(3,4)$ .

Compute the unit normal vector $\mathbf{n}$ and find the reflection matrix $R=\mathbf{I}_2-2\mathbf{n}\mathbf{n}^\top$ .
Compute $T(\mathbf{e}_0)$ , $T(\mathbf{e}_1)$ .
Verify that $\|T(\mathbf{e}_0)\|=\|T(\mathbf{e}_1)\|=1$ and $T(\mathbf{e}_0)\cdot T(\mathbf{e}_1)=0$ .
Let $\mathbf{v}=(-1,1)$ . Prove that $\|\mathbf{v}\|=\|T(\mathbf{v})\|$ and $\mathbf{v}\cdot\mathbf{u}=T(\mathbf{v})\cdot\mathbf{u}$ ( $\mathbf{u}$ is a direction vector of $L$ ).
Compute and compare $\mathbf{N}$ and $T(\mathbf{N})$ , and $\mathbf{u}$ and $T(\mathbf{u})$ ; explain the geometric meaning.
Compute $\det(R)$ and explain its geometric meaning.

Hint. The reflection matrix is $=\mathbf{I}-2\mathbf{n}\mathbf{n}^\top$ ; the reflection reverses the normal vector and leaves the direction vector unchanged.

Exercise 36 (Reflection across

y=x

)

In $\mathbb{R}^2$ , find the matrix of the reflection across the line $y=x$ . Does the direction of the unit normal vector affect the result?

Hint. The normal vector of $y=x$ can be taken to be $(1,-1)$ or $(-1,1)$ ; both give the same $\mathbf{n}\mathbf{n}^\top$ .

Exercise 37 (The General Reflection Formula)

In $\mathbb{R}^2$ , let $L$ be the line through the origin with direction vector $(\cos\theta,\sin\theta)$ .

Explain why the matrix of the reflection across $L$ is $\begin{pmatrix}\cos2\theta&\sin2\theta\\\sin2\theta&-\cos2\theta\end{pmatrix}$ .

Hint. The normal vector is $(-\sin\theta,\cos\theta)$ ; simplify using the double-angle formulas.

Exercise 38 (Properties of Reflection Matrices in

\mathbb{R}^2

)

Let $T$ be the reflection in $\mathbb{R}^2$ across some line through the origin, and let $R$ be its reflection matrix.

Prove that $\|T(\mathbf{e}_0)\|=\|T(\mathbf{e}_1)\|=1$ and $T(\mathbf{e}_0)\cdot T(\mathbf{e}_1)=0$ .
Verify that $R^\top=R$ and $R^2=\mathbf{I}_2$ .
Use (2) to explain why $R^\top R=R R^\top=\mathbf{I}_2$ .
Use (3) to explain why $R^{-1}=R^\top=R$ .
For any two nonzero vectors $\mathbf{v}$ , $\mathbf{u}$ , prove that $\|T(\mathbf{v})\|=\|\mathbf{v}\|$ and $\mathbf{v}\cdot\mathbf{u}=T(\mathbf{v})\cdot T(\mathbf{u})$ .
Compute $\det(R)$ and explain its geometric meaning.

Hint. Use $R=\mathbf{I}-2\mathbf{n}\mathbf{n}^\top$ together with $\|\mathbf{n}\|=1$ .

Exercise 39 (Finding a Reflection Matrix)

Let $\mathbf{u}=(15,20)\in\mathbb{R}^2$ . Decide whether there is a reflection matrix $R$ that maps $\mathbf{u}$ to the given vector; if there is, write down $R$ or the axis of reflection.

To $(24,10)$ .
To $(20,15)$ .
To $(-20,-15)$ .
To $(24,7)$ .

Hint. A reflection preserves length ( $\|R\mathbf{u}\|=\|\mathbf{u}\|$ ), and the axis of reflection lies along the angle bisector of $\mathbf{u}$ and the target vector.

Exercise 40 (Two Reflections Compose to a Rotation)

In $\mathbb{R}^2$ , prove or explain: the composition of the reflections across two different lines is a rotation. If the angle between the two lines is $\alpha$ , the angle of rotation is $2\alpha$ .

Hint. Let the direction angles of the two lines be $\phi_0$ , $\phi_1$ , and compute the product using the general reflection formula $R_{2\phi}$ .

Exercise 41 (Reflection across a Plane in

\mathbb{R}^3

)

In $\mathbb{R}^3$ , let $T$ be the reflection across the plane $x+2y+2z=0$ , with matrix $R$ .

Find the unit normal vector $\mathbf{n}$ , and use $R=\mathbf{I}-2\mathbf{n}\mathbf{n}^\top$ to find the reflection matrix.
Compute $T(1,2,2)$ and explain its geometric meaning.
When $a+2b+2c=0$ , prove that $T(a,b,c)=(a,b,c)$ , and explain its geometric meaning.
Verify that $\|T(\mathbf{e}_i)\|=1$ and $T(\mathbf{e}_i)\cdot T(\mathbf{e}_j)=0$ ( $i\neq j$ ).
Verify that $R^\top=R$ and $R^2=\mathbf{I}_3$ .
Use (5) to explain why $R^{-1}=R^\top=R$ .
Verify that $\det(R)=-1$ , and explain its geometric meaning.

Hint. $(1,2,2)$ is the direction of the normal vector, so it should be reversed by the reflection.

Exercise 42 (Reflection across a Line in Space (Advanced))

In $\mathbb{R}^3$ , let $L$ be the line through the origin with unit direction vector $\mathbf{u}$ . Define the reflection matrix $R=2\mathbf{u}\mathbf{u}^\top-\mathbf{I}$ .

Explain the geometric meaning of this definition.
Use Rodrigues’ formula to write down the rotation matrix $S$ with axis of rotation $\mathbf{u}$ and angle of rotation $\pi$ .
Using the discussion of three-dimensional rotation matrices, prove that $R=S$ .

Hint. The reflection across a line is the same as the rotation by $180°$ about that line.

# ============================================================
print_header("Reflection Mappings")
# ============================================================

print_step(1, "Exercise 35: reflection matrix in ℝ² for the normal vector (3,4)")
N = np.array([3.0, 4]); n = N / np.linalg.norm(N)
R = reflect_mat_2d(n)
print("  R ="); print(R)
compare_print("‖T(e₀)‖", f"{np.linalg.norm(R[:,0]):.4f}", "1 (length-preserving)")
compare_print("T(e₀)·T(e₁) (should be 0)", f"{np.dot(R[:,0], R[:,1]):.2e}", "0 (angle-preserving)")
compare_print("det(R)", f"{np.linalg.det(R):.4f}", "−1 (reflection reverses orientation)")
v = np.array([-1.0, 1])
compare_print("‖v‖=‖T(v)‖", np.isclose(np.linalg.norm(v), np.linalg.norm(R@v)), "True")

print_step(2, "Exercise 36: reflection matrix for the line y=x")
n_yx = np.array([1.0, -1]) / np.sqrt(2)
R_yx = reflect_mat_2d(n_yx)
print("  R_{y=x} ="); print(R_yx)
compare_print("R_{y=x}(1,0) = (0,1)?", R_yx @ np.array([1,0]), np.array([0,1]))

print_step(3, "Exercise 39: finding reflection matrices (mapping u=(15,20) to each target)")
u_vec = np.array([15.0, 20])
targets = [np.array([24.0,10]), np.array([20.0,15]),
           np.array([-20.0,-15]), np.array([24.0,7])]
for i, t in enumerate(targets):
    if np.isclose(np.linalg.norm(t), np.linalg.norm(u_vec)):
        axis_dir = u_vec + t; axis_dir = axis_dir / np.linalg.norm(axis_dir)
        n_axis = np.array([-axis_dir[1], axis_dir[0]])
        R_t = reflect_mat_2d(n_axis)
        print(f"  ({i+1}) R·u =", R_t @ u_vec,
              " target:", t, " ✓" if np.allclose(R_t@u_vec, t) else " ✗")
    else:
        print(f"  ({i+1}) ‖target‖≠‖u‖, no reflection matrix exists")

print_step(4, "Exercise 41: reflection across the plane x+2y+2z=0 in ℝ³")
n3 = np.array([1.0, 2, 2]) / 3
R3 = reflect_mat_3d(n3)
print("  R3 ="); print(np.round(R3, 4))
compare_print("T(1,2,2) (the normal vector is mapped to its negative)", R3 @ np.array([1.0,2,2]),
    "−(1,2,2)")
compare_print("T(2,0,−1) (vectors in the plane are fixed)", R3 @ np.array([2.0,0,-1]),
    np.array([2.0,0,-1]))
compare_print("det(R3)", f"{np.linalg.det(R3):.4f}", "−1")
compare_print("R3² = I", np.allclose(R3@R3, np.eye(3)), "True (a reflection is self-inverse, i.e., an involution)")

print_step(5, "Exercise 42(3): reflection across a line in ℝ³ = Rodrigues(u,π)")
u = np.array([1.0, 0, 0])
R_line = 2*np.outer(u,u) - np.eye(3)
R_rod_pi = rodrigues(u, np.pi)
compare_print("R_line = Rodrigues(u,π)?", np.allclose(R_line, R_rod_pi), "True")

Rotation Mappings¶

Exercise 43 (The Rotation by

\pi/3

)

Consider the rotation mapping $T$ that rotates counterclockwise by $\frac{\pi}{3}$ about the origin, with matrix $R_{\pi/3}$ .

Write down $R_{\pi/3}$ .
Compute $T(\mathbf{e}_0)$ , $T(\mathbf{e}_1)$ .
Verify that $\|T(\mathbf{e}_0)\|=\|T(\mathbf{e}_1)\|=1$ and $T(\mathbf{e}_0)\cdot T(\mathbf{e}_1)=0$ .
Let $\mathbf{v}=(-1,1)$ . Find $T(\mathbf{v})$ , and verify that $\|\mathbf{v}\|=\|T(\mathbf{v})\|$ and that the angle between them is $\frac{\pi}{3}$ .

Hint. $\cos(\pi/3)=1/2$ , $\sin(\pi/3)=\sqrt{3}/2$ .

Exercise 44 (General Properties of Rotation Matrices)

Let $R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$ , with corresponding mapping $T_\theta:\mathbb{R}^2\to\mathbb{R}^2$ .

Prove by computation that $\|T_\theta(\mathbf{e}_0)\|=\|T_\theta(\mathbf{e}_1)\|=1$ and $T_\theta(\mathbf{e}_0)\cdot T_\theta(\mathbf{e}_1)=0$ .
Use (1) to explain why $R_\theta^\top R_\theta=R_\theta R_\theta^\top=\mathbf{I}_2$ .
Verify that $R_\theta^{-1}=R_\theta^\top=R_{-\theta}$ .
For any nonzero vector $\mathbf{v}=(a,b)$ , prove that $\|T_\theta(\mathbf{v})\|=\|\mathbf{v}\|$ and that the (ordinary) angle between them is $\arccos(\cos\theta)$ , while the directed angle of rotation is $\theta$ (modulo $2\pi$ ).
Compute $\det(R_\theta)$ and explain its geometric meaning.
Verify that $R_\alpha R_\beta=R_{\alpha+\beta}$ .

Hint. Use (1) to explain why the columns of $R_\theta$ form an orthonormal basis, and deduce (2) from this.

Exercise 45 (Finding a Rotation Matrix)

Let $\mathbf{u}=(1,2)\in\mathbb{R}^2$ . Decide whether there is a rotation matrix $R$ that maps $\mathbf{u}$ to the given vector; if there is, write down $R$ .

Rotate $\mathbf{u}$ to $(0,4)$ .
Rotate $\mathbf{u}$ to $(2,-1)$ .
Rotate $\mathbf{u}$ to $(2,1)$ .

Hint. Rotations preserve length; if the length of the target vector is $\neq\|\mathbf{u}\|$ , no such rotation exists.

Exercise 46 (Rodrigues’ Rotation Formula)

Take the axis of rotation $\mathbf{u}=(1,1,1)/\sqrt{3}$ . Rodrigues’ formula: $R_{(\mathbf{u},\theta)}=\mathbf{I}_3+(\sin\theta)\mathbf{K}+(1-\cos\theta)\mathbf{K}^2$ .

Write down the skew-symmetric matrix $\mathbf{K}$ and compute $\mathbf{K}^2$ .
Compute the rotation matrix for the angle of rotation $\frac{\pi}{2}$ and verify that $\det=1$ .
Verify that $\|T(\mathbf{e}_i)\|=1$ and $T(\mathbf{e}_i)\cdot T(\mathbf{e}_j)=0$ ( $i\neq j$ ).
Verify that $R^\top R=R R^\top=\mathbf{I}_3$ .
Let $\mathbf{v}=(1,-1,0)$ . Explain why $\mathbf{v}\perp\mathbf{u}$ , and compute the angle between $\mathbf{v}$ and $T(\mathbf{v})$ .
Set $\theta=\pi$ and redo (2)(3)(4)(5).

Hint. Normalize $\mathbf{u}$ , then substitute into the formula.

Exercise 47 (Rodrigues (Continued): Angles Relative to the Axis of Rotation)

Let $\mathbf{u}=(1,1,1)/\sqrt{3}$ and $\mathbf{w}=(1,0,0)$ .

Is the angle between $\mathbf{w}$ and $R_{(\mathbf{u},\pi/2)}\mathbf{w}$ equal to $\pi/2$ ? Does this contradict what a rotation by $\pi/2$ is supposed to mean?

Hint. The angle of rotation $\theta$ is measured in the plane perpendicular to the axis of rotation; it is not the angle through which an arbitrary vector turns.

Exercise 48 (Properties of General Three-Dimensional Rotation Matrices)

Consider the general three-dimensional rotation matrix $R_{(\mathbf{u},\theta)}=\mathbf{I}_3+(\sin\theta)\mathbf{K}+(1-\cos\theta)\mathbf{K}^2$ .

Compute and prove that $\mathbf{K}^2=\mathbf{P}_\mathbf{u}-\mathbf{I}_3$ ( $\mathbf{P}_\mathbf{u}$ is the projection matrix onto the direction of $\mathbf{u}$ ) and that $\mathbf{K}^4=-\mathbf{K}^2$ .
Use (1) and matrix computation to prove that $R^\top R=R R^\top=\mathbf{I}_3$ .
Explain why $R^{-1}=R^\top=R_{(\mathbf{u},-\theta)}$ , and explain its geometric meaning.
Explain why $R_{(-\mathbf{u},-\theta)}=R_{(\mathbf{u},\theta)}$ , and explain its geometric meaning.
For any vector $\mathbf{v}$ , explain why $\|R_{(\mathbf{u},\theta)}\mathbf{v}\|=\|\mathbf{v}\|$ .
For any two nonzero vectors $\mathbf{v}$ , $\mathbf{w}$ , explain why a rotation preserves the angle between them.

Hint. Use $\mathbf{v}\cdot\mathbf{w}=\mathbf{v}^\top\mathbf{w}$ and $\|R\mathbf{v}\|^2=\mathbf{v}^\top R^\top R\mathbf{v}$ .

Exercise 49 (Finding the Axis and Angle of a Rotation from Images)

In three-dimensional space, a rotation $R$ maps $\mathbf{e}_0\mapsto\frac{1}{3}(2,2,1)$ and $\mathbf{e}_1\mapsto\frac{1}{3}(-1,2,-2)$ .

Determine the matrix $R$ (hint: $\det R=1$ ).
Determine the axis and the angle of rotation, and check them with Rodrigues’ formula.

Hint. The axis of rotation is perpendicular to $\mathbf{v}-R\mathbf{v}$ (for every $\mathbf{v}$ ).

# ============================================================
print_header("Rotation Mappings")
# ============================================================

print_step(1, "Exercise 43: rotation by π/3 — the matrix and isometry")
theta = np.pi/3
R = np.array([[np.cos(theta), -np.sin(theta)],
              [np.sin(theta),  np.cos(theta)]])
print("  R_{π/3} ="); print(R)
v = np.array([-1.0, 1])
Tv = R @ v
compare_print("‖v‖ = ‖T(v)‖", np.isclose(np.linalg.norm(v), np.linalg.norm(Tv)), "True (rotations preserve length)")
angle = np.arccos(np.dot(v,Tv)/(np.linalg.norm(v)*np.linalg.norm(Tv)))
compare_print("angle between v and T(v)", f"{angle:.6f}", f"π/3 = {np.pi/3:.6f}")

print_step(2, "Exercise 46: Rodrigues rotation (axis (1,1,1)/√3, θ=π/2)")
u_ax = np.array([1,1,1]) / np.sqrt(3)
R_rod = rodrigues(u_ax, np.pi/2)
print("  R ="); print(np.round(R_rod, 4))
compare_print("det(R)", f"{np.linalg.det(R_rod):.4f}", "1 (rotation matrix)")
compare_print("R^T R = I", np.allclose(R_rod.T @ R_rod, np.eye(3)), "True (orthogonal matrix)")
v = np.array([1.0, -1, 0])
compare_print("v ⊥ axis of rotation", np.isclose(np.dot(v, u_ax), 0), "True")
Tv = R_rod @ v
angle = np.arccos(np.clip(np.dot(v,Tv)/(np.linalg.norm(v)*np.linalg.norm(Tv)), -1, 1))
compare_print("angle between v and T(v)", f"{angle:.4f}", f"π/2 = {np.pi/2:.4f}")

print_step(3, "Exercise 49: rebuild the matrix from the images and find the angle and axis of rotation")
Re0 = np.array([2,2,1])/3; Re1 = np.array([-1,2,-2])/3
Re2 = np.cross(Re0, Re1)
R3 = np.column_stack([Re0, Re1, Re2])
print("  R3 ="); print(np.round(R3, 4))
compare_print("det(R3)", f"{np.linalg.det(R3):.4f}", "1")

trace = np.trace(R3)
theta_found = np.arccos((trace - 1) / 2)
compare_print("angle of rotation θ", f"{theta_found:.4f}", f"π/3 = {np.pi/3:.4f}")

eigvals, eigvecs = np.linalg.eig(R3)
axis_idx = np.argmin(np.abs(eigvals.real - 1.0))
axis = eigvecs[:, axis_idx].real
axis = axis / np.linalg.norm(axis)
print("  axis of rotation =", np.round(axis, 4))

Special Topic: Complex Numbers and Matrices¶

Basic Exercises¶

Exercise 50 (Basics of the Vector Representation of Complex Numbers)

Represent the following complex numbers as two-dimensional vectors:
- $z_0 = 3 + 4i$
- $z_1 = -2 + i$
- $z_2 = 5i$
- $z_3 = -3$
Plot the points corresponding to these complex numbers in the complex plane, and compute their moduli.
Verify the vector nature of complex addition: compute the vector representation of $z_0 + z_1$ and compare it with the result of vector addition.

# ============================================================
print_header("Basics of the Vector Representation of Complex Numbers")
# ============================================================

zs = [3+4j, -2+1j, 5j, -3+0j]
labels = ['z_0 = 3+4i', 'z_1 = -2+i', 'z_2 = 5i', 'z_3 = -3']

print_step(1, "Complex numbers → 2D vectors")
for lbl, z in zip(labels, zs):
    print(f"  {lbl}  →  ({z.real:+.1f}, {z.imag:+.1f})  |modulus| = {abs(z):.4f}")

print_step(2, "Verify complex addition = vector addition")
z0, z1 = zs[0], zs[1]
sum_complex = z0 + z1
v0 = np.array([z0.real, z0.imag])
v1 = np.array([z1.real, z1.imag])
sum_vec = v0 + v1
compare_print("z0+z1 (complex)", f"({sum_complex.real}+{sum_complex.imag}i)",
              f"vector sum {sum_vec}")

print_step(3, "Plot in the complex plane")
fig, ax = plt.subplots(figsize=(5, 5))
colors = ['#57068C', '#006385', '#2AD2C9', '#E37222']
for z, lbl, c in zip(zs, ['$z_0$','$z_1$','$z_2$','$z_3$'], colors):
    ax.annotate('', xy=(z.real, z.imag), xytext=(0,0),
                arrowprops=dict(arrowstyle='->', color=c, lw=2))
    ax.text(z.real+0.1, z.imag+0.1, lbl, color=c, fontsize=12)
ax.axhline(0, color='gray', lw=0.8); ax.axvline(0, color='gray', lw=0.8)
ax.set_xlim(-4, 5); ax.set_ylim(-1, 6)
ax.set_xlabel('Re'); ax.set_ylabel('Im')
ax.set_title('Complex numbers in the complex plane', fontsize=13)
ax.set_aspect('equal'); ax.grid(True, alpha=0.3)
plt.tight_layout(); plt.show()

# ============================================================
print_header("The Rotation Property of the Imaginary Unit")
# ============================================================

Mi = M(1j)   # M(i)

print_step(1, "M(i) acting on the standard basis")
e0 = np.array([1, 0]); e1 = np.array([0, 1])
print(f"  M(i)·e_0 = {Mi @ e0}  (expected [0, 1])")
print(f"  M(i)·e_1 = {Mi @ e1}  (expected [-1, 0])")

print_step(2, "Powers of M(i)")
for k in range(1, 5):
    Mk = np.linalg.matrix_power(Mi, k)
    print(f"  M(i)^{k} =\n{Mk}\n  (corresponds to i^{k} = {1j**k:.0f})")

print_step(3, "Multiplying points on the unit circle by i")
thetas = np.linspace(0, 2*np.pi, 8, endpoint=False)
for th in thetas[:3]:
    v = np.array([np.cos(th), np.sin(th)])
    v_rot = Mi @ v
    compare_print(f"θ={np.degrees(th):.0f}°",
                  np.round(v_rot, 4),
                  f"[cos{np.degrees(th)+90:.0f}°, sin{np.degrees(th)+90:.0f}°] = {np.round([np.cos(th+np.pi/2), np.sin(th+np.pi/2)],4)}")

Exercise 51 (Exploring the Matrix Representation of Complex Numbers)

Using the definition $M(z) = M(a+bi) = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ , write down the matrix representations of the following complex numbers:
- $M(1+i)$
- $M(2-3i)$
- $M(\sqrt{3}+i)$
For $z = 1+i$ :
- Compute the action of $M(1+i)$ on the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$
- Draw the vector before and after the mapping in a coordinate system
- Analyze the geometric meaning of this mapping (angle of rotation and scale factor)
Compute the polar form of the complex number $1+i$ , and verify the geometric effect of the matrix mapping.

# ============================================================
print_header("Exploring the Matrix Representation of Complex Numbers")
# ============================================================

print_step(1, "Write down the matrix representations")
zs3 = [1+1j, 2-3j, np.sqrt(3)+1j]
names = ['M(1+i)', 'M(2-3i)', 'M(√3+i)']
for name, z in zip(names, zs3):
    print(f"  {name} =\n{M(z)}\n")

print_step(2, "The action of M(1+i) on e_0")
Mz = M(1+1j)
e0 = np.array([1., 0.])
result = Mz @ e0
print(f"  M(1+i) · e_0 = {result}")
print(f"  length of the output vector = {np.linalg.norm(result):.4f}  (expected √2 ≈ {np.sqrt(2):.4f})")
print(f"  angle of the output vector = {np.degrees(np.arctan2(result[1], result[0])):.1f}°  (expected 45°)")

print_step(3, "Polar-form check")
z = 1+1j
print(f"  |1+i| = {abs(z):.4f}  (≈ √2 = {np.sqrt(2):.4f})")
print(f"  arg(1+i) = {np.degrees(np.angle(z)):.1f}°")

Exercise 52 (Verifying Complex Multiplication with Matrices)

Theoretical verification. For arbitrary complex numbers $z_0 = a_0 + b_0 i$ and $z_1 = a_1 + b_1 i$ , prove that

M(z_0 z_1) = M(z_0) M(z_1)

(100)

Numerical verification. Take $z_0 = 1+i$ , $z_1 = 2-i$ , compute $M(z_0 z_1)$ and $M(z_0)M(z_1)$ , and verify that they are equal.
Geometric explanation. Explain why it makes geometric sense that complex multiplication corresponds to matrix multiplication.

# ============================================================
print_header("Verifying Complex Multiplication with Matrices")
# ============================================================

z0, z1 = 1+1j, 2-1j

print_step(1, "Compute M(z0·z1) and M(z0)·M(z1)")
product_complex = z0 * z1
Mprod = M(product_complex)
Mmul  = M(z0) @ M(z1)
print(f"  z0·z1 = {product_complex}")
compare_print("M(z0·z1)", Mprod, "M(z0)·M(z1)")
print(f"\n  M(z0)·M(z1) =\n{Mmul}")
print(f"\n  equal? {np.allclose(Mprod, Mmul)}")

print_step(2, "Geometric check: modulus and argument")
print(f"  |z0|={abs(z0):.4f}, arg(z0)={np.degrees(np.angle(z0)):.1f}°")
print(f"  |z1|={abs(z1):.4f}, arg(z1)={np.degrees(np.angle(z1)):.1f}°")
print(f"  |z0·z1|={abs(product_complex):.4f}  (expected {abs(z0)*abs(z1):.4f})")
print(f"  arg(z0·z1)={np.degrees(np.angle(product_complex)):.1f}°  (expected {np.degrees(np.angle(z0)+np.angle(z1)):.1f}°)")

Exercise 53 (Decomposition into Rotation and Scaling)

For a complex number $z = r(\cos\theta + i\sin\theta)$ , prove that

M(z) = r \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}

(105)

Analyze how the matrix $M(z)$ combines scaling (by a factor of $r$ ) and rotation (by the angle $\theta$ ).
For $z = 2e^{i\pi/3}$ :
- Write down its matrix representation
- Analyze its effect on unit vectors
- Verify that the mapping preserves angle relationships

# ============================================================
print_header("Decomposition into Rotation and Scaling")
# ============================================================

z = 2 * np.exp(1j * np.pi / 3)  # 2e^{iπ/3}

print_step(1, "The matrix representation M(z)")
Mz = M(z)
print(f"  z = 2·e^{{iπ/3}} = {z:.4f}")
print(f"  M(z) =\n{Mz}")

print_step(2, "Action on the standard basis")
e0 = np.array([1., 0.])
result = Mz @ e0
print(f"  M(z)·e_0 = {np.round(result, 4)}")
print(f"  length = {np.linalg.norm(result):.4f}  (expected 2.0)")
print(f"  angle = {np.degrees(np.arctan2(result[1], result[0])):.1f}°  (expected 60°)")

print_step(3, "Comparison with the rotation + scaling decomposition")
r, theta = abs(z), np.angle(z)
R = np.array([[np.cos(theta), -np.sin(theta)],
              [np.sin(theta),  np.cos(theta)]])
compare_print("r·R_θ", r * R, "M(z)")
print(f"  equal? {np.allclose(Mz, r * R)}")

Exercise 54 (The Matrix Representation of the Complex Conjugate)

If $z = a + bi$ , what is the matrix representation of its complex conjugate $\bar{z} = a - bi$ ?
Prove that $M(\bar{z}) = M(z)^\top$ (the transpose).
Verify that $M(z)M(\bar{z}) = |z|^2 \mathbf{I}$ , where $\mathbf{I}$ is the identity matrix.
Explain the geometric meaning of this result: why does $z \cdot \bar{z}$ correspond to a multiple of the identity matrix?

# ============================================================
print_header("The Matrix Representation of the Complex Conjugate")
# ============================================================

z = 3 + 4j
zbar = z.conjugate()

print_step(1, "M(z) and M(z̄)")
print(f"  M(z) =\n{M(z)}")
print(f"  M(z̄) =\n{M(zbar)}")
compare_print("M(z̄) vs M(z)^T", M(zbar), M(z).T)
print(f"  M(z̄) == M(z)^T? {np.allclose(M(zbar), M(z).T)}")

print_step(2, "M(z)·M(z̄) = |z|²·I")
product = M(z) @ M(zbar)
expected = abs(z)**2 * np.eye(2)
compare_print("M(z)·M(z̄)", product, f"{abs(z)**2:.0f}·I")
print(f"  equal? {np.allclose(product, expected)}")

print_step(3, "Geometric check: the arguments cancel")
print(f"  arg(z) = {np.degrees(np.angle(z)):.2f}°")
print(f"  arg(z̄) = {np.degrees(np.angle(zbar)):.2f}°  (they cancel → total 0°)")
print(f"  |z|·|z̄| = {abs(z)*abs(zbar):.4f}  (= |z|² = {abs(z)**2:.4f})")

Exercise 55 (Matrix Realizations of Complex Functions)

Complex division.
- For $w = \dfrac{z_0}{z_1}$ ( $z_1 \neq 0$ ), derive an expression for $M(w)$
- Verify that $M(w) = M(z_0)M(z_1)^{-1}$
Powers of complex numbers.
- For $z^n$ , prove that $M(z^n) = M(z)^n$
- Use this result to compute the matrix representation of $(1+i)^8$
A complex equation.
- Find all solutions of the equation $z^2 = -1$
- Verify these solutions with matrix methods

# ============================================================
print_header("Matrix Realizations of Complex Functions")
# ============================================================

print_step(1, "Checking complex division")
z0, z1 = 3+2j, 1-1j
w = z0 / z1
Mw_direct = M(w)
Mw_matrix = M(z0) @ np.linalg.inv(M(z1))
compare_print("M(z0/z1) (directly)", Mw_direct, "M(z0)·M(z1)^{-1}")
print(f"  M(z0)·M(z1)^{{-1}} =\n{np.round(Mw_matrix, 4)}")
print(f"  equal? {np.allclose(Mw_direct, Mw_matrix)}")

print_step(2, "Powers of complex numbers: (1+i)^8")
z = 1+1j
for n in [2, 4, 8]:
    Mpow_direct = M(z**n)
    Mpow_matrix = np.linalg.matrix_power(M(z), n)
    ok = np.allclose(Mpow_direct, Mpow_matrix)
    print(f"  n={n}: M(z^n)==M(z)^n? {ok}  z^n={z**n:.2f}")

print_step(3, "Matrix check of z²=-1")
for z_sol, name in [(1j, 'i'), (-1j, '-i')]:
    mat = np.linalg.matrix_power(M(z_sol), 2)
    print(f"  M({name})² =\n{mat}  (should be -I? {np.allclose(mat, -np.eye(2))})")

Exercise 56 (Matrix Analysis of Mappings of the Complex Plane)

Linear mappings. Analyze the matrix mappings corresponding to the following complex functions:
- $f(z) = (1+i)z$
- $f(z) = iz + 1$ (note: this is not linear!)
- $f(z) = \bar{z}$
Combining geometric mappings. First rotate the complex plane by $45°$ about the origin, then enlarge it by a factor of $\sqrt{2}$ . Express this mapping both by complex multiplication and by matrix multiplication, and verify that the two methods give the same result.
Fixed-point analysis. Find the fixed points of the mapping $f(z) = (1+i)z$ , and analyze them with matrix methods.

# ============================================================
print_header("Matrix Analysis of Mappings of the Complex Plane")
# ============================================================

print_step(1, "Matrices of the three mappings")
# f(z) = (1+i)z
print("  matrix of f(z) = (1+i)z:")
print(M(1+1j))

# f(z) = z̄  (reflection)
M_conj = np.array([[1, 0], [0, -1]])
print("\n  matrix of f(z) = z̄ (reflection):")
print(M_conj)

# f(z) = iz + 1 is affine
print("\n  f(z) = iz+1 is an affine mapping; matrix of its linear part:")
print(M(1j))

print_step(2, "Rotate by 45°, then enlarge by √2: two methods")
# complex multiplication
z_op = np.sqrt(2) * np.exp(1j * np.pi / 4)  # = 1+i
print(f"  complex multiplier = {z_op:.4f}  (≈ 1+i = {1+1j})")
M_complex = M(z_op)
# composition of matrices
theta = np.pi / 4
R45 = np.array([[np.cos(theta), -np.sin(theta)],
                [np.sin(theta),  np.cos(theta)]])
M_matrix = np.sqrt(2) * R45
compare_print("M(1+i) (complex method)", M_complex, "√2·R₄₅° (matrix method)")
print(f"  √2·R₄₅° =\n{np.round(M_matrix, 4)}")
print(f"  equal? {np.allclose(M_complex, M_matrix)}")

print_step(3, "Fixed-point analysis")
A = M(1+1j) - np.eye(2)
print(f"  M(1+i) - I =\n{A}")
print(f"  det(M(1+i) - I) = {np.linalg.det(A):.4f}  (nonzero → unique fixed point 0)")

Special Topic: Skew-Symmetric Matrices and the Cross Product¶

Basic Exercises¶

Exercise 57 (The Correspondence between Vectors and Skew-Symmetric Matrices)

Given a three-dimensional vector $\mathbf{v} = (a, b, c)$ , define the corresponding $3 \times 3$ matrix

[\mathbf{v}]_\times = \begin{bmatrix} 0 & -c & b \\ c & 0 & -a \\ -b & a & 0 \end{bmatrix}

(114)

Part 1. Write down the matrices corresponding to the following vectors:

$\mathbf{u}_0 = (1, 0, 0)$
$\mathbf{u}_1 = (0, 1, 0)$
$\mathbf{u}_2 = (0, 0, 1)$
$\mathbf{v} = (2, -1, 3)$

Part 2. Verify the following properties of these matrices:

Compute $[\mathbf{v}]_\times + [\mathbf{v}]_\times^\top$ . What do you find?
Compute $[\mathbf{u}_0 + \mathbf{u}_1]_\times$ and $[\mathbf{u}_0]_\times + [\mathbf{u}_1]_\times$ , and compare the results
Compute $[2\mathbf{v}]_\times$ and $2[\mathbf{v}]_\times$ , and compare the results

# ============================================================
print_header("The Correspondence between Vectors and Skew-Symmetric Matrices")
# ============================================================

u0 = np.array([1., 0., 0.])
u1 = np.array([0., 1., 0.])
u2 = np.array([0., 0., 1.])
v  = np.array([2., -1., 3.])

print_step(1, "Compute the skew-symmetric matrix of each vector")
for name, vec in [('u0', u0), ('u1', u1), ('u2', u2), ('v=(2,-1,3)', v)]:
    print(f"  [{name}]× =\n{skew(vec)}\n")

print_step(2, "Verify skew-symmetry: [v]× + [v]×^T = 0")
Sv = skew(v)
result = Sv + Sv.T
print(f"  [v]× + [v]×^T =\n{result}")
print(f"  zero matrix? {np.allclose(result, 0)}")

print_step(3, "Verify linearity")
Su0_Su1 = skew(u0) + skew(u1)
S_u0_u1 = skew(u0 + u1)
print(f"  [u0+u1]× == [u0]×+[u1]×? {np.allclose(S_u0_u1, Su0_Su1)}")
compare_print("[2v]× vs 2[v]×", skew(2*v), 2*skew(v))
print(f"  [2v]× == 2[v]×? {np.allclose(skew(2*v), 2*skew(v))}")

Exercise 58 (The Relation between Matrix-Vector Multiplication and the Cross Product)

Part 1. Verification by concrete computation

Take $\mathbf{u} = (1, 0, 0)$ , $\mathbf{v} = (0, 1, 0)$ : compute $[\mathbf{u}]_\times \mathbf{v}$ and $\mathbf{u} \times \mathbf{v}$ , and compare the results.
Take $\mathbf{u} = (1, 2, 3)$ , $\mathbf{v} = (4, 5, 6)$ : repeat the computation and comparison above.
Take $\mathbf{u} = (2, -1, 3)$ , $\mathbf{v} = (1, 1, -2)$ : repeat the computation and comparison above.

Part 2. General verification

Let $\mathbf{u} = (u_0, u_1, u_2)$ , $\mathbf{v} = (v_0, v_1, v_2)$ . Prove that $[\mathbf{u}]_\times \mathbf{v} = \mathbf{u} \times \mathbf{v}$ .

# ============================================================
print_header("The Relation between Matrix-Vector Multiplication and the Cross Product")
# ============================================================

pairs = [
    (np.array([1.,0.,0.]), np.array([0.,1.,0.])),
    (np.array([1.,2.,3.]), np.array([4.,5.,6.])),
    (np.array([2.,-1.,3.]), np.array([1.,1.,-2.])),
]

for i, (u, v) in enumerate(pairs):
    print_step(i+1, f"u={u}, v={v}")
    mat_result = skew(u) @ v
    cross_result = np.cross(u, v)
    print(f"  [u]×·v = {mat_result}")
    print(f"  u × v  = {cross_result}")
    print(f"  equal? {np.allclose(mat_result, cross_result)}")

Exercise 59 (The Geometric Meaning of Skew-Symmetric Matrices)

Background. We know that $[\mathbf{u}]_\times \mathbf{v} = \mathbf{u} \times \mathbf{v}$ .

Part 1. Verifying orthogonality

For an arbitrary vector $\mathbf{u}$ , compute $[\mathbf{u}]_\times \mathbf{u}$ . What is the result?
Explain why this is the result (hint: properties of the cross product)
Where does the matrix $[\mathbf{u}]_\times$ send the vector $\mathbf{u}$ ?

Part 2. Analyzing the linear transformation

Let $\mathbf{u} = (0, 0, 1)$ (the unit vector along the $z$ axis).

Write down the matrix $[\mathbf{u}]_\times$
Compute the images of the following vectors under this transformation: $\mathbf{e}_0, \mathbf{e}_1, \mathbf{e}_2$
Describe the geometric effect of this linear transformation

Part 3. The connection with rotation

Compare the results above with the effect of “rotation by $90°$ about the $z$ axis”
Compute the determinant of $[\mathbf{u}]_\times$ . What does the result tell you?

# ============================================================
print_header("The Geometric Meaning of Skew-Symmetric Matrices")
# ============================================================

u = np.array([0., 0., 1.])
Su = skew(u)

print_step(1, "[u]×·u = 0 (null space)")
print(f"  [u]×·u = {Su @ u}  (should be the zero vector)")

print_step(2, "Images of the standard basis")
for i, e in enumerate([np.eye(3)[:,j] for j in range(3)]):
    print(f"  [u]×·e_{i} = {Su @ e}")

print_step(3, "Determinant and comparison")
det_Su = np.linalg.det(Su)
print(f"  det([u]×) = {det_Su:.4f}  (singular matrix; the mapping is not invertible)")

# compare with the matrix of the rotation by 90° about the z axis
Rz90 = np.array([[0,-1,0],[1,0,0],[0,0,1]], dtype=float)
print(f"\n  rotation by 90° about the z axis:\n{Rz90}")
print(f"  [u]×:\n{Su}")
print(f"  they differ in entry (2,2): the rotation matrix keeps the z axis, while [u]× collapses it")

Exercise 60 (Matrix Commutators and the Cross Product)

Definition. The commutator of two matrices $\mathbf{A}$ , $\mathbf{B}$ is $[\mathbf{A}, \mathbf{B}] = \mathbf{AB} - \mathbf{BA}$ .

Part 1. A concrete computation

Take $\mathbf{u} = (1, 0, 0)$ , $\mathbf{v} = (0, 1, 0)$ : compute $[[\mathbf{u}]_\times, [\mathbf{v}]_\times]$ and $[\mathbf{u} \times \mathbf{v}]_\times$ , and compare the results.

Part 2. Verification with another example

Take $\mathbf{u} = (1, 1, 0)$ , $\mathbf{v} = (0, 1, 1)$ and repeat the computation.

Part 3. Spotting the pattern

Based on your computations, conjecture whether the following relation holds:

[[\mathbf{u}]_\times, [\mathbf{v}]_\times] = [\mathbf{u} \times \mathbf{v}]_\times

(123)

# ============================================================
print_header("Matrix Commutators and the Cross Product")
# ============================================================

def commutator(A, B):
    return A @ B - B @ A

pairs = [
    (np.array([1.,0.,0.]), np.array([0.,1.,0.])),
    (np.array([1.,1.,0.]), np.array([0.,1.,1.])),
]

for i, (u, v) in enumerate(pairs):
    print_step(i+1, f"u={u}, v={v}")
    Su, Sv = skew(u), skew(v)
    comm = commutator(Su, Sv)
    cross_skew = skew(np.cross(u, v))
    print(f"  [[u]×,[v]×] =\n{comm}")
    print(f"  [u×v]× =\n{cross_skew}")
    print(f"  equal? {np.allclose(comm, cross_skew)}")

Exercise 61 (◆ Approximating Small-Angle Rotations (Rodrigues’ Formula))

Background. Rodrigues’ rotation formula gives a general expression for a rotation about an arbitrary axis:

\mathbf{R} = \mathbf{I} + (\sin\theta)\mathbf{K} + (1-\cos\theta)\mathbf{K}^2

(128)

where $\mathbf{K} = [\mathbf{u}]_\times$ is the skew-symmetric matrix corresponding to the unit axis of rotation $\mathbf{u}$ .

Part 1. Verify the formula with the example $\mathbf{u} = (0,0,1)$ (rotation about the $z$ axis), $\theta = 90°$ .

Part 2. Derive the small-angle approximations (first and second order).

Part 3. For the rotation by $\theta = 60°$ about the axis $\mathbf{u} = \frac{1}{\sqrt{3}}(1,1,1)$ , compute the exact rotation matrix and apply this rotation to $\mathbf{v} = (1,0,0)$ .

Part 4. Verify the key properties (invariance of the axis, orthogonality, determinant).

Part 5. For the angular velocity $\boldsymbol{\omega} = (0,0,2)$ rad/s and the initial condition $\mathbf{R}(0)=\mathbf{I}$ , verify the differential equation and approximate its solution numerically.

Solution to Exercise 61

Step 1 (Part 1). $\mathbf{u} = (0,0,1)$ , $\mathbf{K} = \begin{bmatrix}0&-1&0\\1&0&0\\0&0&0\end{bmatrix}$ :

\mathbf{K}^2 = \begin{bmatrix}-1&0&0\\0&-1&0\\0&0&0\end{bmatrix}

(129)

Substituting $\theta = 90°$ :

\mathbf{R} = \mathbf{I} + \mathbf{K} + \mathbf{K}^2 = \begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix}

(130)

This is the standard matrix of the rotation by $90°$ about the $z$ axis, so the formula checks out.

Step 2 (small-angle approximation). Below, $\theta$ is in radians with $\lvert\theta\rvert\ll1$ , and the axis $\mathbf{u}$ is fixed. Substituting $\sin\theta=\theta+O(\theta^3)$ and $1-\cos\theta=\theta^2/2+O(\theta^4)$ into Rodrigues’ formula gives

First order: $\mathbf{R}=\mathbf{I}+\theta\mathbf{K}+O(\theta^2)$ .
Second order: $\mathbf{R}=\mathbf{I}+\theta\mathbf{K}+\dfrac{\theta^2}{2}\mathbf{K}^2+O(\theta^3)$ .

The matrix remainder terms are understood with respect to any fixed matrix norm.

Step 3 (Part 3). With $\mathbf{u}=\frac{1}{\sqrt3}(1,1,1)$ and $\theta=60°=\pi/3$ , substituting $\sin\theta=\sqrt3/2$ and $\cos\theta=1/2$ gives

\mathbf{R}=\frac13\begin{pmatrix}2&-1&2\\2&2&-1\\-1&2&2\end{pmatrix},\qquad \mathbf{R}\begin{pmatrix}1\\0\\0\end{pmatrix}=\frac13\begin{pmatrix}2\\2\\-1\end{pmatrix}.

(131)

Step 4 (key properties). For a unit axis $\mathbf{u}$ , $\mathbf{K}\mathbf{u}=0$ , so $\mathbf{R}\mathbf{u}=\mathbf{u}$ . The vector triple product gives $\mathbf{K}^2=\mathbf{u}\mathbf{u}^\top-\mathbf{I}$ , hence $\mathbf{K}^3=-\mathbf{K}$ and $\mathbf{K}^4=-\mathbf{K}^2$ . Let $s=\sin\theta$ , $q=1-\cos\theta$ , and expand using $\mathbf{K}^\top=-\mathbf{K}$ :

\mathbf{R}^\top\mathbf{R}=(\mathbf{I}-s\mathbf{K}+q\mathbf{K}^2)(\mathbf{I}+s\mathbf{K}+q\mathbf{K}^2)=\mathbf{I}+(2q-s^2-q^2)\mathbf{K}^2=\mathbf{I}.

(132)

Hence $\det\mathbf{R}\in\{-1,1\}$ . Since $\mathbf{R}(\theta)$ depends continuously on $\theta$ and $\mathbf{R}(0)=\mathbf{I}$ , the determinant cannot jump between the two values, so $\det\mathbf{R}=1$ for all $\theta$ . $\square$

Step 5 (the differential equation). Fix the angular velocity $\boldsymbol{\omega}=(0,0,2)$ rad/s and let $\mathbf{W}=[\boldsymbol{\omega}]_\times$ . The general solution of $d\mathbf{R}/dt=\mathbf{W}\mathbf{R}$ is $\mathbf{R}(t)=e^{\mathbf{W}t}\mathbf{R}(0)$ ; starting from the initial attitude $\mathbf{R}(0)=\mathbf{I}$ , we get $\mathbf{R}(t)=e^{\mathbf{W}t}=\begin{pmatrix}\cos(2t)&-\sin(2t)&0\\\sin(2t)&\cos(2t)&0\\0&0&1\end{pmatrix}$ .

$\underline{\text{Rodrigues' formula }\mathbf{R} = \mathbf{I} + \sin\theta\,\mathbf{K} + (1-\cos\theta)\mathbf{K}^2 \text{ is equivalent to the matrix exponential }e^{\theta\mathbf{K}}\text{, as the Taylor expansion of the latter shows.}}$

# ============================================================
print_header("Rodrigues' Rotation Formula")
# ============================================================

def rodrigues(u, theta):
    """Rotation by theta radians about the unit axis u (Rodrigues' formula)"""
    K = skew(u / np.linalg.norm(u))
    return np.eye(3) + np.sin(theta)*K + (1-np.cos(theta))*(K@K)

print_step(1, "Check: rotation by 90° about the z axis")
u_z = np.array([0., 0., 1.])
K = skew(u_z)
K2 = K @ K
R90 = rodrigues(u_z, np.pi/2)
print(f"  K² =\n{K2}")
print(f"  R(z,90°) =\n{np.round(R90, 4)}")
standard_Rz90 = np.array([[0,-1,0],[1,0,0],[0,0,1]], float)
print(f"  = standard matrix of the rotation by 90° about the z axis? {np.allclose(R90, standard_Rz90)}")

print_step(2, "Small-angle approximation (θ=0.1 rad)")
theta_s = 0.1
R_exact  = rodrigues(u_z, theta_s)
R_order1 = np.eye(3) + theta_s * K
R_order2 = np.eye(3) + theta_s * K + (theta_s**2/2) * K2
print(f"  exact:\n{np.round(R_exact, 6)}")
print(f"  first-order approximation error: {np.max(np.abs(R_exact - R_order1)):.2e}")
print(f"  second-order approximation error: {np.max(np.abs(R_exact - R_order2)):.2e}")

print_step(3, "Rotation by 60° about the axis (1,1,1)/√3")
u_111 = np.ones(3) / np.sqrt(3)
R60 = rodrigues(u_111, np.pi/3)
print(f"  R =\n{np.round(R60, 4)}")
v = np.array([1., 0., 0.])
print(f"  R·v = {np.round(R60 @ v, 4)}")

print_step(4, "Verify the key properties")
print(f"  R·u = {np.round(R60 @ u_111, 4)}  (should be ≈ u = {np.round(u_111,4)})")
print(f"  R^T·R = I? {np.allclose(R60.T @ R60, np.eye(3))}")
print(f"  det(R) = {np.linalg.det(R60):.6f}  (should be 1)")

print_step(5, "Numerical check of the differential equation (ω=(0,0,2), Δt=0.01)")
omega = np.array([0., 0., 2.])
Kw = skew(omega)
dt = 0.01
R0 = np.eye(3)
R_approx = R0 + dt * Kw @ R0
R_exact_dt = rodrigues(u_z, 2 * dt)   # rotation about the z axis by ω·dt = 2*0.01 rad
print(f"  error of the approximate solution: {np.max(np.abs(R_approx - R_exact_dt)):.2e}")

Determinants¶

Exercise 62 (Basic Determinant Computations)

Compute the following determinants (make good use of the properties of determinants):

$\begin{vmatrix}3&5\\2&7\end{vmatrix}$
$\begin{vmatrix}-4&6\\2&-3\end{vmatrix}$
$\begin{vmatrix}-3&0\\79&4\end{vmatrix}$
$\begin{vmatrix}1&3&-9\\0&4&23\\0&0&9\end{vmatrix}$
$\begin{vmatrix}3&-7&9\\1&-1&3\\-2&2&6\end{vmatrix}$
$\begin{vmatrix}1&2&3\\6&6&6\\7&8&9\end{vmatrix}$

Hint. For item 4, use the fact that an upper triangular determinant $=$ the product of the diagonal entries; for item 6, check whether row 1 is a linear combination of the other rows.

Exercise 63 (

\det(3A)

and

3\det(A)

)

Let $\mathbf{A}=\begin{pmatrix}1&4\\2&5\end{pmatrix}$ .

Compute $\det(\mathbf{A})$ and $\det(3\mathbf{A})$ .
Is $\det(3\mathbf{A})=3\det(\mathbf{A})$ ?

Hint. For an $n\times n$ matrix, $\det(c\mathbf{A})=c^n\det(\mathbf{A})$ .

Exercise 64 (

\det(\mathbf{A}\mathbf{B})=\det(\mathbf{A})\det(\mathbf{B})

)

Let $\mathbf{A}=\begin{pmatrix}2&1\\3&-1\end{pmatrix}$ , $\mathbf{B}=\begin{pmatrix}1&4\\0&2\end{pmatrix}$ .

Compute $\det(\mathbf{A})$ , $\det(\mathbf{B})$ , $\det(\mathbf{A}\mathbf{B})$ .
Verify that $\det(\mathbf{A}\mathbf{B})=\det(\mathbf{A})\cdot\det(\mathbf{B})$ .

Hint. First compute $\mathbf{A}\mathbf{B}$ , then take its determinant.

Exercise 65 (

\det(\mathbf{C})=\det(\mathbf{C}^\top)

)

Let $\mathbf{C}=\begin{pmatrix}1&0&2\\-1&3&1\\2&1&0\end{pmatrix}$ .

Compute $\det(\mathbf{C})$ and $\det(\mathbf{C}^\top)$ .
Verify that $\det(\mathbf{C})=\det(\mathbf{C}^\top)$ .

Hint. That transposing a matrix leaves its determinant unchanged is a basic property of determinants.

Exercise 66 (The Area of a Parallelogram)

Use a determinant to compute the area of the parallelogram spanned by the vectors $\mathbf{u}=(1,3)$ and $\mathbf{v}=(4,-2)$ . If $\mathbf{u}$ , $\mathbf{v}$ are replaced by $8\mathbf{u}$ and $5\mathbf{v}$ , what is the new area?

Hint. The area is $=|\det(\mathbf{u},\mathbf{v})|$ ; use the behavior of determinants under scalar multiples.

Exercise 67 (The Volume of a Parallelepiped)

In $\mathbb{R}^3$ , consider the three vectors $\mathbf{u}=(2,0,0)$ , $\mathbf{v}=(0,4,0)$ , $\mathbf{w}=(0,0,1)$ .

Compute the volume of the parallelepiped spanned by these three vectors, and describe its geometric shape.

Hint. The volume is $=|\det(\mathbf{u},\mathbf{v},\mathbf{w})|$ ; all three vectors lie along the axes.

Exercise 68 (Linear Transformations and Area (the Unit Square))

The matrix $\mathbf{A}=\begin{pmatrix}2&1\\0&3\end{pmatrix}$ determines the linear transformation $T_A:\mathbb{R}^2\to\mathbb{R}^2$ .

Consider the unit square with vertices $(a,b),(a+1,b),(a+1,b+1),(a,b+1)$ .

The images of the four vertices under $T$ determine a quadrilateral $\Sigma$ . What are its shape and area?
Compute $\det(\mathbf{A})$ and compare it with (1).

Hint. A linear transformation maps parallelograms to parallelograms, and the area scale factor is $=|\det(\mathbf{A})|$ .

Exercise 69 (Linear Transformations and Area (a General Parallelogram))

The matrix $\mathbf{A}=\begin{pmatrix}0&3\\2&1\end{pmatrix}$ determines the linear transformation $T_A$ .

The vectors $\mathbf{u}=(a,b)$ , $\mathbf{v}=(c,d)$ span the parallelogram $\Omega$ , and the image of $\Omega$ under $T$ is $\Sigma$ .

Is $\Sigma$ a parallelogram?
How are the area of $\Omega$ , the area of $\Sigma$ , and $\det(\mathbf{A})$ related?

Hint. Linear transformations preserve the structure of parallelograms.

Exercise 70 (The Geometric Meaning of the Determinant)

In $\mathbb{R}^2$ , let $\mathbf{u}=(a,b)$ , $\mathbf{v}=(c,d)$ , and let $\Sigma$ be the parallelogram they span.

If $\mathbf{u}\perp\mathbf{v}$ , is $\det(\mathbf{u},\mathbf{v})=\|\mathbf{u}\|\|\mathbf{v}\|$ ?
Explain geometrically: which $\mathbf{v}$ make $\det(\mathbf{u},\mathbf{v})=\|\mathbf{u}\|\|\mathbf{v}\|$ ? Which make $\det(\mathbf{u},\mathbf{v})=-\|\mathbf{u}\|\|\mathbf{v}\|$ ?

Hint. $\det(\mathbf{u},\mathbf{v})=\|\mathbf{u}\|\|\mathbf{v}\|\sin\theta$ , where $\theta$ is the directed angle from $\mathbf{u}$ to $\mathbf{v}$ .

Exercise 71 (The Cross Product and the Scalar Triple Product)

In $\mathbb{R}^3$ , let $\mathbf{a}=(1,2,0)$ , $\mathbf{b}=(3,0,4)$ , $\mathbf{c}=(5,6,0)$ .

Compute $\mathbf{b}\times\mathbf{c}$ and $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})$ .
Compute the determinant $\det(\mathbf{a},\mathbf{b},\mathbf{c})$ and compare it with (1).

Hint. The scalar triple product satisfies $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=\det(\mathbf{a},\mathbf{b},\mathbf{c})$ .

# ============================================================
print_header("Determinants")
# ============================================================

print_step(1, "Exercise 62: basic determinant computations")
mats = [
    np.array([[3,5],[2,7]]),
    np.array([[-4,6],[2,-3]]),
    np.array([[-3,0],[79,4]]),
    np.array([[1,3,-9],[0,4,23],[0,0,9]]),
    np.array([[3,-7,9],[1,-1,3],[-2,2,6]]),
    np.array([[1,2,3],[6,6,6],[7,8,9]]),
]
for i, m in enumerate(mats):
    print(f"  det_{i+1} = {np.linalg.det(m):.4f}")

print_step(2, "Exercise 63: det(3A) vs 3·det(A) (scale factor = cⁿ)")
A = np.array([[1,4],[2,5]])
compare_print("det(3A)", f"{np.linalg.det(3*A):.4f}",
    f"3²·det(A) = {9*np.linalg.det(A):.4f}")

print_step(3, "Exercise 64: determinant of a product, det(AB) = det(A)·det(B)")
A = np.array([[2,1],[3,-1]]); B = np.array([[1,4],[0,2]])
compare_print("det(AB)", f"{np.linalg.det(A@B):.4f}",
    f"det(A)·det(B) = {np.linalg.det(A)*np.linalg.det(B):.4f}")

print_step(4, "Exercise 65: det(C) = det(C^T)")
C = np.array([[1,0,2],[-1,3,1],[2,1,0]])
compare_print("det(C) vs det(C^T)",
    f"{np.linalg.det(C):.4f}", f"{np.linalg.det(C.T):.4f}")

print_step(5, "Exercise 66: area of a parallelogram = |det[u,v]|")
u, v = np.array([1,3]), np.array([4,-2])
area = abs(np.linalg.det(np.column_stack([u,v])))
print(f"  original area = {area:.4f}")
compare_print("area spanned by 8u, 5v", 40*area, "40 × original area")

print_step(6, "Exercise 67: volume of a parallelepiped = |det[a,b,c]|")
volume_matrix = np.column_stack([np.array([2,0,0]), np.array([0,4,0]), np.array([0,0,1])])
compare_print("volume |det|", f"{abs(np.linalg.det(volume_matrix)):.4f}", "8 (axis-aligned cuboid)")

print_step(7, "Exercise 71: cross product and scalar triple product, a·(b×c) = det[a,b,c]")
b, c = np.array([3,0,4]), np.array([5,6,0])
bxc = np.cross(b, c)
a = np.array([1,2,0])
volume_matrix = np.column_stack([a,b,c])
print("  b×c =", bxc)
compare_print("a·(b×c) vs det[a,b,c]",
    f"{np.dot(a,bxc):.4f}", f"{np.linalg.det(volume_matrix):.4f}")

Systems of Linear Equations¶

Exercise 72 (Solving a System of Two Equations (Full Steps))

Solve the following system of linear equations:

\begin{cases}2x+y=5\\x+y=3\end{cases}

(144)

Write it in the coefficient-matrix form $\mathbf{A}\mathbf{x}=\mathbf{b}$ .
Compute $\det(\mathbf{A})$ and confirm that $\mathbf{A}$ is invertible.
Find $\mathbf{A}^{-1}$ .
Solve using $\mathbf{x}=\mathbf{A}^{-1}\mathbf{b}$ .

Hint. The inverse formula for a matrix of order 2: $\mathbf{A}^{-1}=\frac{1}{\det(\mathbf{A})}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$ .

Exercise 74 (Solving a System of Three Equations (Full Steps))

Solve: $\begin{cases}x+y+z=6\\x+2y+z=8\\x+y+2z=9\end{cases}$

Write it in the form $\mathbf{A}\mathbf{x}=\mathbf{b}$ .
Compute $\det(\mathbf{A})$ and confirm that it is invertible.
Find $\mathbf{A}^{-1}$ .
Solve using $\mathbf{x}=\mathbf{A}^{-1}\mathbf{b}$ .

Hint. Use the formula from the text to find $\mathbf{A}^{-1}$ (elimination on the augmented matrix).

Exercise 75 (Solving a System of Three Equations (Inverse-Matrix Method))

Solve: $\begin{cases}2x+y=5\\x+y+z=5\\y+3z=4\end{cases}$

Hint. Use $\mathbf{x}=\mathbf{A}^{-1}\mathbf{b}$ .

# ============================================================
print_header("Systems of Linear Equations")
# ============================================================

print_step(1, "Exercise 72: 2×2 system (elimination)")
A = np.array([[2,1],[1,1]]); b = np.array([5,3])
x = np.linalg.solve(A, b)
print("  solution x =", x)
compare_print("check Ax", A@x, b)

print_step(2, "Exercise 73: 2×2 system (inverse-matrix method)")
A = np.array([[3,-1],[1,2]]); b = np.array([7,4])
x = np.linalg.solve(A, b)
print("  solution x =", x)
compare_print("check Ax", A@x, b)

print_step(3, "Exercise 74: 3×3 system (with the inverse matrix)")
A = np.array([[1,1,1],[1,2,1],[1,1,2]]); b = np.array([6,8,9])
x = np.linalg.solve(A, b)
print("  solution x =", x)
compare_print("check Ax", A@x, b)
print("  A⁻¹ ="); print(np.linalg.inv(A))

print_step(4, "Exercise 75: 3×3 system (inverse-matrix method)")
A = np.array([[2,1,0],[1,1,1],[0,1,3]]); b = np.array([5,5,4])
x = np.linalg.solve(A, b)
print("  solution x =", x)
compare_print("check Ax", A@x, b)