Linear Maps

Definitions and interpretation
First-order approximation of non-linear maps

Definition and Intepretation

Definition

A map $f: mathbf{R}^n rightarrow mathbf{R}^m$ is linear (resp. affine) if and only if every one of its components is. The formal definition we saw here for functions applies verbatim to maps.

To an

matrix

, we can associate a linear map $f : mathbf{R}^n rightarrow mathbf{R}^m$ , with values f(x) = Ax

. Conversely, to any linear map, we can uniquely associate a matrix

which satisfies f(x) = Ax

for every

Indeed, if the components of

, are linear, then they can be expressed as f_i(x) = a_i^Tx

for some $a_i in mathbf{R}^n$ . The matrix

is the matrix that has a_i^T

as its

-th row:

$f(x) = left(begin{array}{c} f_1(x) vdots f_n(x) end{array}right) = left(begin{array}{c} a_1^Tx vdots a_n^Tx end{array}right) = Ax, ;; mbox{ with } A := left(begin{array}{c} a_1^T vdots a_m^T end{array}right) in mathbf{R}^{m times n}.$

Hence, there is a one-to-one correspondence between matrices and linear maps. This is extending what we saw for vectors, which are in one-to-one correspondence with linear functions.

Representation of affine maps via the matrix-vector product. A function $f: mathbf{R}^n rightarrow mathbf{R}^m$ is affine if and only if it can be expressed via a matrix-vector product:

for some unique pair (A,b)

, with $A in mathbf{R}^{m times n}$ and $b in mathbf{R}^m$ . The function is linear if and only if b = 0

The result above shows that a matrix can be seen as a (linear) map from the ‘‘input“ space $mathbf{R}^n$ to the ‘‘output” space $mathbf{R}^m$ . Both points of view (matrices as simple collections of vectors, or as linear maps) are useful.

Interpretations