Linear Maps and Matrices

Abstract

Let

$$A = \left( {\begin{array}{*{20}{c}} {{{a}_{{11}}}\quad \cdots \quad {{a}_{{1n}}}} \\ { \vdots \quad \quad \quad \quad \vdots } \\ {{{a}_{{m1}}}\quad \cdots \quad {{a}_{{mn}}}} \\ \end{array} } \right)$$

be an m × n matrix. We can then associate with A a map

$${L_A}:{K^n} \to {K^m}$$

by letting

$${L_A}(X) = AX$$

for every column vector X in K ⁿ. Thus L _A is defined by the association X ↦ AX, the product being the product of matrices. That L _A is linear is simply a special case of Theorem 3.1, Chapter II, namely the theorem concerning properties of multiplication of matrices. Indeed, we have (math) for all vectors X, Y in K ⁿ and all numbers c. We call L _A the linear map associated with the matrix A.