MathH54hw5 (1)

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

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Unformatted text preview: 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 { 1 ,..., 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 { 1 ,..., p , 1 ,..., m } for W 1 , and it may be extended to a basis { 1 ,..., p , 1 ,..., n } for W 2 . By the definition of W 1 + W 2 it is easy to see that (2) { 1 ,..., p ,...
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This note was uploaded on 02/22/2012 for the course EE 441 taught by Professor Neely during the Spring '08 term at USC.

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

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