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Assignment 3

# Assignment 3 - MATH 223 Linear Algebra Winter 2010...

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MATH 223, Linear Algebra Winter, 2010 Assignment 3, due in class Monday, February 1, 2010 1. Let V = M 2 ( R ) be the real vector space of 2 × 2 matrices with real entries. For each of the following subsets of V , decide if it is independent, if it is a spanning set for V , and/or if it is a basis for V . Justify your answers. (a) { 1 3 2 - 1 , 2 0 5 0 , 4 1 0 - 1 } . (b) { 1 1 1 1 , 1 1 - 1 - 1 , 1 - 1 1 - 1 , 1 2 3 5 } . (c) { 1 0 0 0 , 1 1 0 0 , 1 1 1 0 , 1 1 1 1 , 0 1 1 1 } . 2. Find a basis for each of the null space, row space and column space of the following matrix over C . A = 1 2 - i - 3 + 2 i 3 i 0 0 - 1 + i 2 - 2 i - 2 - 3 + i 1 + i 2 - 3 + i - 3 + i - 1 - 3 i 2 i 3 + 6 i - 6 - 10 i - 7 + 3 i 6 + 5 i . Express each row of A as a linear combination of the vectors in your basis for the row space, and express each column of A as a linear combination of the vectors in your basis for the column space. 3. In this problem we suppose that F is a field, A is an m × n matrix over F and that W is a subspace of F m . (a) Show that U = { ~v F n : A~v W } is a subspace of F n . (b) Now suppose that m = n and A is invertible, and that B = { ~v 1 , . . . ,~v k } is a basis for W . Show that { A - 1 ~v 1 , . . . , A - 1 ~v k } is a basis for U
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