Problem I Let A denote the matrix given below. (i) Find the rank and the nullity of the matrix A. (ii) Find a basis for the row space of A. (iii) Find a basis for the column space of A, then extend it to a basis for R4. 1 1. 2 2 1-20 1 -1 20 -1 1-2 0 1 20-1 -1 2. 1 -1 1 -1 -2 -1 2 1 -1 1 -2 2 1 -2 10 -1 2 -1 0 1-2 1 2-1 2 -1
Problem I Let A denote the matrix given below. (i) Find the rank and the nullity of the matrix A. (ii) Find a basis for the row space of A. (iii) Find a basis for the column space of A, then extend it to a basis for R4. 1 1. 2 2 1-20 1 -1 20 -1 1-2 0 1 20-1 -1 2. 1 -1 1 -1 -2 -1 2 1 -1 1 -2 2 1 -2 10 -1 2 -1 0 1-2 1 2-1 2 -1
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.5: Subspaces, Basis, Dimension, And Rank
Problem 63EQ
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Transcribed Image Text:Problem I Let A denote the matrix given below.
(i) Find the rank and the nullity of the matrix A.
(ii) Find a basis for the row space of A.
(iii) Find a basis for the column space of A, then extend it to a basis for R4.
1
1.
4.
7.
10.
13.
16.
19.
1
1 -1
2
1
2
-1
1
-2 10
1
2 -1 0
-1
2 1 -2 10
3
-1
2 -1
1
2
1 -2 0 1
20 -1
-20 1
20 -1
1
1
1
-1
2 1 -1
1 -2 2 1
22 -1
-1
-2 1
-1 1 2
1 1 -2 1
1 2-2 1
2
3 -1 3 2 -1
10 1 -2
20-1 2 -1
20 1 -2
1 0 -1 2
1 1 2
1 -1 1 -2 1
2-1 2-2 1
1 2 -1
1
1
2
2-2
1
2
1 2 1
14 -1
2 -1 2 1
2 -1 4 1
2.
5.
8.
11.
14.
17.
20.
1
1 -2 -1
1
-1
2 1 -2 -1
-1 2 1
1
1
1-2 1
1 -1 2 -1 1
2 1 -2 1
-1
3
-1
2 -1
123
----
1
-1
1
-1
1 -1
1
1
1
-2
-1
2 1 -1
1
1
7
1 -2 2-1
-3-2 1
1 1
2 -1
2 1 -2 -2
1 -1
2 -1
1
2-2
-2 -1
2-1
-1
-2 -2 1
2-2 -1
777
-1 -2 1
-1 2 -1
-2 -1
-1
-1 -2 -2
-1 -2 -2 -4
1 -1 2 -2
2-2 -1
1
-1
-1 -1
1
-1 -2 2
2 1
4 -1
2
2-1 -4 1
3.
6.
9.
12.
15.
18.
10
1-2
-1 2 -1 0
-2 10
2 -1 0
2 1
2
-1
1
1 -1
2
3
-2 0 1
20-1
1 -2 0 1
2 0 -1
-1
1-2 1 2
-1 2 1 2
1
1
2 1 -2
2
14
-1 2 -1 4
1
1
1
-1
33-1
1
2-2 1
22 1 -2 -1
2 2-1
1
2222 2442
2 -1
-1
2 -1 2 1
1 -1
1
-1 -1
2 -1 -2
-1
1
1
4-1
1
1
-22
22
24
22
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