Linear Maps and Matrices

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


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


Vector Space Vector Space Versus Clockwise Rotation Coordinate Vector Invertible Matrix 
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Copyright information

© Springer Science+Business Media New York 1987

Authors and Affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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