Linear Algebra pp 81-94 | Cite as

# Linear Maps and Matrices

Chapter

## Abstract

Let be an by letting for every column vector

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

*m*×*n*matrix. We can then associate with*A*a map$${L_A}:{K^n} \to {K^m}$$

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

*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*.## 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.

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## Copyright information

© Springer Science+Business Media New York 1987