(a) i. a basis for Row(A). What is the dimension of Row(A)? ii. a basis for Col(A). What is the dimension of Col(A)? iii. a basis for Nul(A). What is the dimension of Nul(A)? Given the matrices A and the reduced echelon form of A below, find: (b) A = 1234 5 1 0 1 2 -1 2246 4 1120 1 Consider vectors rref (A) = 1 0 1 0 0 1 1 0 0 0 0 1 00000 -5/3 8/3 1/3 V₁ = [1,-1, 5, 2], V2 = [−2, 3, 1, 0], V3 = [9, -10,34, 14], V4 = [0, 4, 2, -3], V5 = [-3, 31, 41, -8] in R4. Find a basis and the dimension of the subspace W = span{V1, V2, V3, V4, V5}. 10 705 0 1 -1 04 00 016 00 000 NOTE: RREF of the matrix whose columns are vectors V₁, V2, V3, V4, V5 is marks) Let A be an m x n matrix. Prove that every vector in the null space of A is orthogonal to every vector in the row space of A.
(a) i. a basis for Row(A). What is the dimension of Row(A)? ii. a basis for Col(A). What is the dimension of Col(A)? iii. a basis for Nul(A). What is the dimension of Nul(A)? Given the matrices A and the reduced echelon form of A below, find: (b) A = 1234 5 1 0 1 2 -1 2246 4 1120 1 Consider vectors rref (A) = 1 0 1 0 0 1 1 0 0 0 0 1 00000 -5/3 8/3 1/3 V₁ = [1,-1, 5, 2], V2 = [−2, 3, 1, 0], V3 = [9, -10,34, 14], V4 = [0, 4, 2, -3], V5 = [-3, 31, 41, -8] in R4. Find a basis and the dimension of the subspace W = span{V1, V2, V3, V4, V5}. 10 705 0 1 -1 04 00 016 00 000 NOTE: RREF of the matrix whose columns are vectors V₁, V2, V3, V4, V5 is marks) Let A be an m x n matrix. Prove that every vector in the null space of A is orthogonal to every vector in the row space of 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
![2.
"!
(a)
(b)
Given the matrices A and the reduced echelon form of A below, find:
i. a basis for Row(A). What is the dimension of Row(A)?
ii. a basis for Col(A). What is the dimension of Col(A)?
iii. a basis for Nul(A). What is the dimension of Nul(A)?
A
1234 5
1012 -1
2246 4
11 20 1
Consider vectors
rref (A) =
0 1 0
0
1 1 0
0 0 0 1
0000
-5/3
8/3
1/3
0
V₁ = [1,-1, 5, 2], V₂ = [-2, 3, 1, 0], V3 = [9, -10, 34, 14], V4 = [0, 4, 2, -3], V5 = [-3, 31, 41, -8]
in R4. Find a basis and the dimension of the subspace W = span{V1, V2, V3, V4, V5}.
10
01
00
00
NOTE: RREF of the matrix whose columns are vectors V₁, V2, V3, V4, V5 is
705
-1 0 4
016
000
marks) Let A be an m x n matrix. Prove that every vector in the null space of A is
orthogonal to every vector in the row space of A.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9ff5cc5-ffa0-4d62-80cc-4b5d73842eb0%2F1ac06658-7469-4266-8622-1bd1bd8b9fe3%2Fuvg9ljj_processed.png&w=3840&q=75)
Transcribed Image Text:2.
"!
(a)
(b)
Given the matrices A and the reduced echelon form of A below, find:
i. a basis for Row(A). What is the dimension of Row(A)?
ii. a basis for Col(A). What is the dimension of Col(A)?
iii. a basis for Nul(A). What is the dimension of Nul(A)?
A
1234 5
1012 -1
2246 4
11 20 1
Consider vectors
rref (A) =
0 1 0
0
1 1 0
0 0 0 1
0000
-5/3
8/3
1/3
0
V₁ = [1,-1, 5, 2], V₂ = [-2, 3, 1, 0], V3 = [9, -10, 34, 14], V4 = [0, 4, 2, -3], V5 = [-3, 31, 41, -8]
in R4. Find a basis and the dimension of the subspace W = span{V1, V2, V3, V4, V5}.
10
01
00
00
NOTE: RREF of the matrix whose columns are vectors V₁, V2, V3, V4, V5 is
705
-1 0 4
016
000
marks) Let A be an m x n matrix. Prove that every vector in the null space of A is
orthogonal to every vector in the row space of A.
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