# Linear Maps and Matrices

• Serge Lang
Part of the Undergraduate Texts in Mathematics book series (UTM)

## 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 XAX, 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.

## Keywords

Vector Space Vector Space Versus Clockwise Rotation Coordinate Vector Invertible Matrix
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.