ConceptLinear Algebra used inMSIAI-written · not yet proofread

Linear Map

Also Linear Transformation · Linear Function · Lineare Abbildung · Homomorphismus

A linear map is a function between vector spaces that preserves addition and scalar multiplication; between ℝⁿ and ℝᵐ every linear map is multiplication by a matrix.

Properties#

Zero is fixed: $f(0) = f(0 \cdot x) = 0 \cdot f(x) = 0$ . A map with $f(0) \neq 0$ is never linear.

Proof

Write $x = \sum_{i=1}^n x_i e_i$ . Linearity gives $f(x) = \sum_{i=1}^n x_i f(e_i)$ , which is the matrix–vector product of $\big(f(e_1) \mid \dots \mid f(e_n)\big)$ with $x$ . Uniqueness: any $A$ with $f(x) = Ax$ has $i$ -th column $Ae_i = f(e_i)$ . The converse follows from $A(\lambda x + \mu y) = \lambda Ax + \mu Ay$ .

Composition is the matrix product: if $f(x) = Ax$ and $g(y) = By$ , then $(g \circ f)(x) = BAx$ . The inverse map, if it exists, is the inverse matrix.
Kernel and image: for $f(x) = Ax$ $f (x) = A x$ with $A \in \mathbb{R}^{m \times n}$ $A \in R^{m \times n}$ :
- $\ker(A) = \{x \in \mathbb{R}^n \mid Ax = 0\}$ , a subspace of $\mathbb{R}^n$ (the inputs that get lost)
- $\operatorname{im}(A) = \{Ax \mid x \in \mathbb{R}^n\}$ , a subspace of $\mathbb{R}^m$ spanned by the columns (the outputs that are reached)
- $\dim \operatorname{im}(A)$ is the rank, and $\operatorname{rank}(A) + \dim\ker(A) = n$ (rank–nullity).
Injective, surjective, bijective:
- injective $\iff \ker(A) = \{0\} \iff \operatorname{rank}(A) = n$
- surjective $\iff \operatorname{im}(A) = \mathbb{R}^m \iff \operatorname{rank}(A) = m$
- bijective $\iff \operatorname{rank}(A) = n = m$ , i.e. $A$ is an invertible matrix

Example#

Example

Map	Linear?	Matrix / reason
Rotation of $\mathbb{R}^2$ by 90°	yes	$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ : $e_1 \mapsto e_2$ , $e_2 \mapsto -e_1$
Projection $(x_1, x_2, x_3) \mapsto (x_1, x_2)$	yes	$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$ , kernel $= \operatorname{span}\{e_3\}$
Derivative $p \mapsto p'$ on polynomials	yes	$(p + q)' = p' + q'$ , $(\lambda p)' = \lambda p'$
$x \mapsto x + 1$	no	$f(0) = 1 \neq 0$
$x \mapsto x^2$	no	$(x + y)^2 \neq x^2 + y^2$ in general

The projection is not injective: $(1, 2, 5)$ and $(1, 2, 7)$ both map to $(1, 2)$ , so it has no inverse.

Sources#

Standard linear algebra. There’s no textbook in the subject’s sources: yet.

Connections

Mentioned in inline links7

Jacobian MatrixANA
Best linear approximation: $f(x + h) = f(x) + J(x)\,h + o(\|h\|)$ . The linear map $h \mapsto J(x)h$ is the derivative of $f$ at $x$ . §
Determinant
Similar matrices $B = S^{-1}AS$ have the same determinant. So $\det$ is a property of the linear map, independent of the basis. §
Eigenvalue
Basis-independent: $A$ and $A^\top$ have the same eigenvalues, and so do similar matrices $S^{-1}AS$ (same $p_A$ ). So the eigenvalues, like $\det$ and $\operatorname{tr}$ , belong to the linear map, not to its matrix (MML p. 106). §
Eigenvector
A linear map usually turns a vector as well as stretching it. Left: the unit vectors $x$ (dots) and their images $Ax$ (arrows) for $A = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix}$ . Almost every arrow leaves the line through its dot. Righ… §
Matrix Rank
i.e. the maximal number of linearly independent columns of $A$ . Here $\operatorname{im}(A) = \{Ax \mid x \in \mathbb{R}^n\}$ is the column space, i.e. the image of the linear map $x \mapsto Ax$ . §
Determinants and Eigenvalues
One picture behind all three. A linear map in $\mathbb{R}^2$ turns the unit square into a parallelogram: §
Rank and Invertibility
Linear Map — what a matrix does: $x \mapsto Ax$ , with kernel (lost inputs) and image (reached outputs) §
Uniqueness vs. existence: full column rank ( $\ker(A) = \{0\}$ , the map is injective) gives uniqueness. Full row rank ( $\operatorname{im}(A) = \mathbb{R}^m$ , the map is surjective) gives existence. A square matrix needs both, but for a squa… §

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