Linear Algebra pp 81-94 | Cite as
Linear Maps and Matrices
Chapter
Abstract
Let be an m × n matrix. We can then associate with A a map by letting 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.
$$A = \left( {\begin{array}{*{20}{c}} {{{a}_{{11}}}\quad \cdots \quad {{a}_{{1n}}}} \\ { \vdots \quad \quad \quad \quad \vdots } \\ {{{a}_{{m1}}}\quad \cdots \quad {{a}_{{mn}}}} \\ \end{array} } \right)$$
$${L_A}:{K^n} \to {K^m}$$
$${L_A}(X) = AX$$
Keywords
Vector Space Vector Space Versus Clockwise Rotation Coordinate Vector Invertible Matrix
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© Springer Science+Business Media New York 1987