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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 5

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online