For the matrix A = R ANTOTT -2 1 300 -1 3 -2 1-3 4 1 1-3 let R be the reduced row echelon form (RREF) of A and R be the RREF of AT. Direct calculations give that -250 -4 10 0 6 5-70-6 [1 0 11 0 3] 0 1 300 00 0 10 00 0 00 00000 00000 3 [1 2 0 2 0 2 0 0 1 1 0 1 and R= 0 0 0 0 1 1 000000 000000 Find the bases for the row space and column space of the matrix A consisting entirely of row vectors, or column vectors, from A. What are the dimensions of the fundamental spaces row(A), col(A), null(A) and null(AT) for the matrix A?
For the matrix A = R ANTOTT -2 1 300 -1 3 -2 1-3 4 1 1-3 let R be the reduced row echelon form (RREF) of A and R be the RREF of AT. Direct calculations give that -250 -4 10 0 6 5-70-6 [1 0 11 0 3] 0 1 300 00 0 10 00 0 00 00000 00000 3 [1 2 0 2 0 2 0 0 1 1 0 1 and R= 0 0 0 0 1 1 000000 000000 Find the bases for the row space and column space of the matrix A consisting entirely of row vectors, or column vectors, from A. What are the dimensions of the fundamental spaces row(A), col(A), null(A) and null(AT) for the matrix A?
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 3CEXP
Related questions
Question
How do we know what row & colums to pick?
![For the matrix
A =
R=
1
2
0
[1 0 11 0 3
0 1 3 0 0
0 0 0 1 0
00000
00000
000 00
-4
3
4
let R be the reduced row echelon form (RREF) of A and R be the RREF of AT. Direct
calculations give that
5
0
3
10 0 6
-70 -6
3
0 0
1-3
-2
1 1-3
2
and R =
7
[1 2 0 2 0 27
0
0 1 1 0 1
0 0 0 11
000000
000000
Find the bases for the row space and column space of the matrix A consisting entirely
of row vectors, or column vectors, from A. What are the dimensions of the fundamental
spaces row(A), col(A), null(A) and null(AT) for the matrix A?
3
.
Solution: From the positions of the leading 1's in the the RREF R of AT, we know
that the basis for the row space row (A) is {r, rar). That is,
{[1 2 5 0 3], [-2 5 -7 0 -6], [-1 3 -2 1-3]}.
From the positions of the leading 1's in the the RREF R of A, we know that the
basis for the column space col(A) is {C₁, C2, C4). That is,
The dimensions of the fundamental spaces of A are: dim(row(A)) = dim (col(A)) =
3, dim(null(A)) = 5-3 = 2 and dim(null(AT)) = 6 -3 = 3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbee182a-544b-4a35-ae8e-c9ecfc559983%2Fdec34ae1-8e77-4f0d-8e76-672f4f93f78e%2Fgu27vas_processed.png&w=3840&q=75)
Transcribed Image Text:For the matrix
A =
R=
1
2
0
[1 0 11 0 3
0 1 3 0 0
0 0 0 1 0
00000
00000
000 00
-4
3
4
let R be the reduced row echelon form (RREF) of A and R be the RREF of AT. Direct
calculations give that
5
0
3
10 0 6
-70 -6
3
0 0
1-3
-2
1 1-3
2
and R =
7
[1 2 0 2 0 27
0
0 1 1 0 1
0 0 0 11
000000
000000
Find the bases for the row space and column space of the matrix A consisting entirely
of row vectors, or column vectors, from A. What are the dimensions of the fundamental
spaces row(A), col(A), null(A) and null(AT) for the matrix A?
3
.
Solution: From the positions of the leading 1's in the the RREF R of AT, we know
that the basis for the row space row (A) is {r, rar). That is,
{[1 2 5 0 3], [-2 5 -7 0 -6], [-1 3 -2 1-3]}.
From the positions of the leading 1's in the the RREF R of A, we know that the
basis for the column space col(A) is {C₁, C2, C4). That is,
The dimensions of the fundamental spaces of A are: dim(row(A)) = dim (col(A)) =
3, dim(null(A)) = 5-3 = 2 and dim(null(AT)) = 6 -3 = 3.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,