d. Suppose that: A = and B = ア。 ア。 for some k e R (note that k could be zero). Let us write 31 = 71 + k • 72. Show that every row in the set S1 set S2 = {31,72, Generalize your argument in part (d) and prove that if A and B have exactly the same rows, but row i of B is: %3D {71,72, ...,7n} is a linear combination of the rows in the 7n}, and vice versa. .... е. 3 = 7, + k •7,, %D then each row of B is a linear combination of the rows of A, and vice-versa. This completes the proof that rowspace(A) = rowspace(B). n 2.3 The Fundamental Matrix Spaces 151 Explain why the non-zero rows of the rref R of A form a basis for the rowspace of A. Be sure to address both issues: Spanning and linear independence. f. に だ ... 15
d. Suppose that: A = and B = ア。 ア。 for some k e R (note that k could be zero). Let us write 31 = 71 + k • 72. Show that every row in the set S1 set S2 = {31,72, Generalize your argument in part (d) and prove that if A and B have exactly the same rows, but row i of B is: %3D {71,72, ...,7n} is a linear combination of the rows in the 7n}, and vice versa. .... е. 3 = 7, + k •7,, %D then each row of B is a linear combination of the rows of A, and vice-versa. This completes the proof that rowspace(A) = rowspace(B). n 2.3 The Fundamental Matrix Spaces 151 Explain why the non-zero rows of the rref R of A form a basis for the rowspace of A. Be sure to address both issues: Spanning and linear independence. f. に だ ... 15
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
2.3 #4
The question is in the picture
Please answer d, e and f
Transcribed Image Text:Proof of Theorem 2.3.1: The goal of this Exercise is to prove that if B is obtained from A
using a single row operation, then rowspace(A) = rowspace(B). We must show that
every row in B is a linear combination of the rows of A, and every row in A is a linear
combination of the rows of B.
4.
Transcribed Image Text:d.
Suppose that:
Ti + k • 72
A
and B =
for some k e R (note that k could be zero). Let us write 31 = 71 + k • 72. Show that
every row in the set S1 = {71,72, ...,7n} is a linear combination of the rows in the
set S2 = {31,72, ..., 7n}, and vice versa.
е.
Generalize your argument in part (d) and prove that if A and B have exactly the
same rows, but row i of B is:
Si = 7, + k•7,
then each row of B is a linear combination of the rows of A, and vice-versa.
This completes the proof that rowspace(A) = rowspace(B).
ection 2.3 The Fundamental Matrix Spaces
151
f.
Explain why the non-zero rows of the rref R of A form a basis for the rowspace of
A. Be sure to address both issues: Spanning and linear independence.
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