Assignment 4 - Solutions

Assignment 4- - MATH 223 Linear Algebra Winter 2010 Assignment 4 due in class Monday February 8 2010 1 Find a basis for each of the row space

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Winter, 2010 Assignment 4, due in class Monday, February 8, 2010 1. Find a basis for each of the row space, column space and null space of the following matrix A over the complex numbers. What is its rank? Express each row of A as a linear combination of the rows in your basis for the row space; express each of the columns of A as a linear combination of the vectors in your basis for the column space. A = 1 2 i 2 + 4 i 1 + 8 i 4 + 14 i 1 - i 3 + 2 i 7 + i 12 + 7 i 23 + 9 i i 0 - 2 - 2 + i - 3 + 2 i 4 - 1 + i 10 i 1 + 11 i 4 + 22 i . Solution: We perform R 2 7→ R 2 + ( - 1 + i ) R 1 , then R 3 7→ R 3 - iR 1 , then R 4 7→ R 4 - 4 R 1 , then R 1 7→ R 1 - 2 iR 2 , then R 3 7→ R 3 - 2 R 2 , then R 4 7→ R 4 + (1 + 7 i ) R 2 , then R 1 7→ R 1 + ( - 2 - 4 i ) R 3 , then R 2 7→ R 2 + ( - 5 + i ) R 3 and finally, R 4 7→ R 4 - 2 iR 3 to produce the RREF matrix 1 0 2 i 1 + 2 i 0 0 1 1 - i 3 0 0 0 0 0 1 0 0 0 0 0 . A basis for the row space is { ~v 1 ,~v 2 ,~v 3 } = { ( 1 0 2 i 1 + 2 i 0 ) , ( 0 1 1 - i 3 0 ) , ( 0 0 0 0 1 ) } . A basis for the column space is { C 1 ,C 2 ,C 5 } = 1 1 - i i 4 , 2 i 3 + 2 i 0 - 1 + i , 4 + 14 i 23 + 9 i - 3 + 2 i 4 + 22 i . The rank r ( A ) is 3. A basis for the null space is - 2 i - 1 + i 1 0 0 , - 1 - 2 i - 3 0 1 0 . If the original rows are relabelled R 1 ,...,R 4 , we have R 1 = 1 ~v 1 + 2 i~v 2 + (4 + 14 i ) ~v 3 , R 2 = (1 - i ) ~v 1 + (3 + 2 i ) ~v 2 + (23 + 9 i ) ~v 3 , R 3 = i~v 1 + 0 ~v 2 + ( - 3 + 2 i ) ~v 3 and R 4 = 4 ~v 1 + ( - 1 + i ) ~v 2 + (4 + 22 i ) ~v 3 . If the original columns are labelled C 1 ,...,C 5 , then of course C 1 = 1 C 1 + 0 C 2 + 0 C 3 and so on, but also C 3 = 2 iC 1 + (1 - i ) C 2 and C 4 = (1 + 2 i ) C 1 + 3 C 2 . 2. Let V = Z 4 2 and W 1 = Span 1 0 1 0 , 0 1 0 1 and W 2 = Span 0 1 1 0 , 1 0 0 1 and be subspaces of V . Find a basis for W 1 + W 2 and one for W 1 W 2 . 1
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This note was uploaded on 02/10/2010 for the course MATH math 223 taught by Professor Loveys during the Winter '10 term at McGill.

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Assignment 4- - MATH 223 Linear Algebra Winter 2010 Assignment 4 due in class Monday February 8 2010 1 Find a basis for each of the row space

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