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
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.
Download
to read the full chapter text
Copyright information
© Springer Science+Business Media New York 1987