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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
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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2.3 #4

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Please answer d, e and f

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: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.
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.
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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