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?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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.
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