Unformatted text preview: E ij ) ij = 1 (the ij entry is 1), and all other entries are 0. Show that E ij E st = E it if j = s and that E ij E st = 0 n × n when j ≠ s . Show that every A ∈ R n x n is a linear combination of the various E ij . J. Show that A ∈ M n ( R ) satisfies tr ( A ) = 0 if and only if A is a linear combination of: { E ij  i ≠ j } ∪ { E jj – E nn  1 ≤ j ≤ n – 1}. K. Given 1 ≤ i < j ≤ n , describe P ∈ M n ( R ) so that for any A ∈ R n x m , B = PA is the matrix A with rows i and j interchanged: row i ( B ) = row j ( A ), row j ( B ) = row i ( A ), and row k ( B ) = row k ( A ) for all k ≠ i , j . Show P has the same effect on the columns of any C ∈ R pxn . Hand In 1A) p.131. #10; 16; 26; 30; B; C; D; Fii) 1B) G; H; J; K Due Wednesday, August 31...
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 '07
 Guralnick
 Math, Differential Equations, Linear Algebra, Algebra, Equations, Diagonal matrix, C. Let Bn

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