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MathH54hw5 (1)

MathH54hw5 (1) - MATH H54 HOMEWORK 5 DUE MONDAY Exercise 1...

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MATH H54, HOMEWORK 5, DUE MONDAY, OCTOBER 10, 2011 Exercise 1. Prove that the space of all m × n matrices with real entries has dimension mn , by exhibiting a basis for this space. Answer. Let E i,j be the matrix with 1 in the i th row and j th column and with 0 in every other position. There are mn such matrices, and taken together they are a basis for the given vector space. (This is “the standard basis” of the vector space of all m × n matrices.) Exercise 2. Prove “the Counting Theorem”: If W 1 and W 2 are finite-dimensional subspaces of a vector space V , then W 1 + W 2 is finite-dimensional and (1) dim W 1 + dim W 2 = dim( W 1 W 2 ) + dim( W 1 + W 2 ) . Answer. Let { α 1 , . . . , α p } be a basis for dim( W 1 W 2 ). By Theorem 11 in Section 4.5 (“the Extension Theorem”), this may be extended to a basis { α 1 , . . . , α p , β 1 , . . . , β m } for W 1 , and it may be extended to a basis { α 1 , . . . , α p , γ 1 , . . . , γ n } for W 2 . By the definition of W 1 + W 2 it is easy to see that (2) { α 1 , . . . , α p , β 1 , . . . , β m , γ 1 , . . . , γ n } spans the subspace W 1 + W 2 . Hence, by the Spanning Set Theorem in Section 4.3,
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