Table of Common Notations - Linear Algebra with Python

This table collects the key concepts, notations, and results used throughout the book, so that you can build a coherent overall picture of linear algebra.

A note on the structure of the book: §2.1 refers to the first section of Chapter 2. Material that is harder than its surroundings is marked with ◆, which means you may skip it on a first reading; material that has been brought forward out of order for the sake of completeness is marked with ◇.

Logic¶

Symbol	Meaning
$\{x \mid P(x)\}$	The set of all $x$ for which the statement $P(x)$ holds (the vertical bar $\mid$ is read “such that”).
$\forall$ , $\exists$	“For all” and “there exists”. For example, $\forall x \in A,\ \exists y \in B$ is read “for every $x$ in $A$ there exists some $y$ in $B$ ”.
$\Rightarrow$	“Implies” (if ... then ...)
$\Leftrightarrow$	“If and only if” (implication in both directions)
$\mid S\mid$	The cardinality of the set $S$ , that is, its number of elements (for a finite set, simply a count).

Sets and Spaces¶

Symbol	Meaning
$\mathbb{K}$	Abstract field (the symbol comes from German Körper)
$\mathbb{R}$	The set of real numbers
$\mathbb{C}$	The set of complex numbers
$\mathbb{R}^n$	$n$ -dimensional real vector space; Cartesian coordinate system
$\mathbb{C}^n$	$n$ -dimensional complex vector space
$V, W$	General vector spaces
$\text{span}_{\mathbb{K}}\{\mathbf{v}_0, \mathbf{v}_1, \ldots, \mathbf{v}_{n-1}\}$	The span of the vectors $\mathbf{v}_0, \mathbf{v}_1, \ldots, \mathbf{v}_{n-1}$ over the field $\mathbb{K}$
$P(A)$	The power set of the set $A$
$\operatorname{card}(A)$	The number of elements in the set $A$
$P_n(\mathbb{R})$	The space of real polynomials of degree at most $n$
$M_{m \times n}(\mathbb{R})$	The space of $m \times n$ real matrices
$S_n(\mathbb{R})$	The space of $n \times n$ real symmetric matrices

Vectors and Matrices¶

Symbol	Meaning
$\mathbf{v}$ or $\mathbf{v}⟩$ or $\vec{v}$	Vector (a column vector by default)
$\overrightarrow{\mathbf{uv}}$	The line segment from the point $\mathbf{u}$ to the point $\mathbf{v}$
$\mathbf{v}^{\top}$	The transpose of a vector
$⟨\mathbf{v}$ or $\mathbf{v}^{*}$	The conjugate transpose of a vector
$\|\mathbf{v}\|$ or $\|\|\mathbf{v}\|\|$	The $l_2$ length (modulus, norm) of a vector
$\|\|\mathbf{v}\|\|_{l_p}$	The $l_p$ norm of a vector
$\mathbf{0}$	Zero vector; zero matrix
$\mathbf{e}_i$	Standard basis vector (its $i$ -th component is 1 and all others are 0)
$\mathbf{A}$	Matrix
$\|\|\mathbf{A}\|\|$	The norm (modulus) of a matrix
$a_{ij}$ or $[\mathbf{A}]_{ij}$ or $(\mathbf{A})_{ij}$	The entry of the matrix in row $i$ and column $j$
$\mathbf{I}_n$	The $n \times n$ identity matrix
$\mathbf{A}^{\top}$	The transpose of the matrix $\mathbf{A}$
$\mathbf{A}^{*}$	The conjugate transpose of the matrix $\mathbf{A}$
$\mathbf{A}^{-1}$	The inverse of the matrix $\mathbf{A}$
$\mathbf{A}^{+}$	The pseudoinverse of the matrix $\mathbf{A}$
$\text{row}_i(\mathbf{A})$	The $i$ -th row of the matrix $\mathbf{A}$
$\text{col}_j(\mathbf{A})$	The $j$ -th column of the matrix $\mathbf{A}$
$\text{Row}(\mathbf{A})$	The row space of the matrix $\mathbf{A}$
$\text{Col}(\mathbf{A})$ or $\text{Im}(\mathbf{A})$	The column space (image) of the matrix $\mathbf{A}$
$\det(\mathbf{A})$ or $\|\mathbf{A}\|$	The determinant of the matrix $\mathbf{A}$
$\text{tr}(\mathbf{A})$	The trace of the matrix $\mathbf{A}$

Operations¶

Symbol	Meaning
$:= , =:$	Definitional equality (the colon marks the side being defined)
$\mathbf{u} \cdot \mathbf{v}$ or $\mathbf{u}^{\top} \mathbf{v}$	Inner product (dot product) of real vectors
$⟨\mathbf{u}\|\mathbf{v}⟩$ or $\mathbf{u}^{*} \mathbf{v}$	Inner product of (complex) vectors
$\mathbf{u} \times \mathbf{v}$	Cross product of vectors, defined only in (two and) three dimensions
$\mathbf{A} \circ \mathbf{B}$	Hadamard product (entrywise multiplication)
$\mathbf{A} \otimes \mathbf{B}$	Kronecker product
$\mathbf{u}\mathbf{v}^*$ or $\|\mathbf{u}\rangle\langle\mathbf{v}\|$	Outer product (a rank-1 matrix)
$\text{Proj}_{\mathbf{v}}(\mathbf{u})$ or $\frac{\|\mathbf{v}\rangle\langle\mathbf{v}\|\mathbf{u}\rangle}{\langle\mathbf{v}\|\mathbf{v}\rangle}$	The projection of the vector $\mathbf{u}$ onto the vector $\mathbf{v}$

Mappings and Subspaces¶

Symbol	Meaning
$f: X \rightarrow Y$	A mapping from the set $X$ to the set $Y$
$T: V \rightarrow W$	A linear transformation from the vector space $V$ to $W$
$T_{\mathbf{A}}: V \rightarrow W$	The linear transformation from $V$ to $W$ represented by the matrix $\mathbf{A}$
$\ker(T)$	The kernel (null space) of the linear transformation $T$
$\text{Im}(T)$	The image of the linear transformation $T$
$f[A]$	The image of the set $A$ under the mapping $f$
$f^{-1}[B]$	The preimage of the set $B$ under the mapping $f$
$\text{rank}(\mathbf{A})$ or $\dim \text{Im}(T_{\mathbf{A}})$ or $\dim \text{Im}(\mathbf{A})$	The rank of the matrix $\mathbf{A}$ (the dimension of its image)
$\text{nullity}(\mathbf{A})$ or $\dim \ker(T_{\mathbf{A}})$ or $\dim \ker(\mathbf{A})$	The nullity of the matrix $\mathbf{A}$ (the dimension of its kernel)

Eigenvalues and Eigenvectors¶

Symbol	Meaning
$\lambda$	Eigenvalue
$\mathbf{v}_\lambda$	The eigenvector corresponding to the eigenvalue $\lambda$
$\mathbf{A}\mathbf{v} = \lambda\mathbf{v}$	The eigenvalue equation
$\det(\mathbf{A}-\lambda\mathbf{I})=0$	The characteristic polynomial equation