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Unformatted text preview: MATRIX ALGEBRA JOS ´ E MALAG ´ ONL ´ OPEZ Matrices are objects that play different roles in the study of vector spaces: they forme one of the most basic examples of vector spaces, the problem of solving linear systems can be stated and solved in terms of matrices, and later we will see that matrices are the “good” functions between vector spaces. A characteristic of matrices that makes us try to express problems in linear algebra in terms of them is that matrices behave like numbers. The goal of this lecture is to establish the basic algebraic properties of matrices. Basic Arithmetic of Matrices Let M m × n denote the vector space of all the m × n matrices. Let A and B two m × n matrices. Recall that we say that A and B are equal if they are equal entrywise. In other words, if A = ( a ij ) and B = ( b ij ), then A = B if and only if a ij = b ij , for any 1 ≤ i ≤ m , 1 ≤ j ≤ n . We defined the sum A + B entrywise: ( a ij ) + ( b ij ) = ( a ij + b ij ). Simi larly, we defined the difference A − B entrywise. An important matrix in M m × n is the zero matrix , denoted as . The basic properties of addition of matrices are summarized below. Theorem 1. Let A , B and C matrices in M m × n for which the following operations are defined. Then (1) A + B = B + A . (2) A + ( B + C ) = ( A + B ) + C . (3) A − A = . (4) A + = + A = A . 1 We also defined scalar product as α ( a ij ) = ( αa ij ). Moreover, we defined the negative of a matrix A as − A := ( − 1) A . The main properties of scalar product are summarized in the following. Theorem 2. Let A , B and C matrices in M m × n for which the following operations are defined. Then (5) 1 A = A ....
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 Fall '10
 JoseMalagonLopez
 Linear Algebra, Algebra, Linear Systems, Matrices, Vector Space

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