3. Matrix Range, Rank, and Pseudo-inverse Let A e Cmxn and b E C. Show that the following three statements are equivalent: - There exists a vector x E C" such that Ax = b, - Rank(A) = Rank ([A b]) where [A b] = Cmx(n+1) is a matrix obtained by appending b after the last column of A, and - AA¹b = b.
3. Matrix Range, Rank, and Pseudo-inverse Let A e Cmxn and b E C. Show that the following three statements are equivalent: - There exists a vector x E C" such that Ax = b, - Rank(A) = Rank ([A b]) where [A b] = Cmx(n+1) is a matrix obtained by appending b after the last column of A, and - AA¹b = b.
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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![3. Matrix Range, Rank, and Pseudo-inverse
Let A E Cmxn and b E Cm. Show that the following three
statements are equivalent:
- There exists a vector x E C" such that Ax = b,
- Rank(A) = Rank([A b]) where [A b] = Cmx(n+1) is a matrix
obtained by appending b after the last column of A, and
AA¹b = b.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f948d5c-51dd-4f43-ade9-5fd5821144d3%2Fa4e6a631-57c8-4b07-b03d-286a5f59b8de%2Fnz5avp9_processed.png&w=3840&q=75)
Transcribed Image Text:3. Matrix Range, Rank, and Pseudo-inverse
Let A E Cmxn and b E Cm. Show that the following three
statements are equivalent:
- There exists a vector x E C" such that Ax = b,
- Rank(A) = Rank([A b]) where [A b] = Cmx(n+1) is a matrix
obtained by appending b after the last column of A, and
AA¹b = b.
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