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6. Prove the following statements (a) Let V and W be finite-dimensional F-vector spaces, and let TV- W be a linear transformation. Then 3 linear transformation S: W → V such that S. T = Iv iff T is onto (b) Let A & Mmxn (F). Then 3 a matrix B € Mnxm (F) 5. t. BA = In iff rank (A) = h

6. Prove the following statements (a) Let V and W be finite-dimensional F-vector spaces, and let TV- W be a linear transformation. Then 3 linear transformation S: W → V such that S. T = Iv iff T is onto (b) Let A & Mmxn (F). Then 3 a matrix B € Mnxm (F) 5. t. BA = In iff rank (A) = h

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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that if I is invertible, then real с numbers there are at most h in vertible. sit. T+cs is not 6. Prove the following statements (a) Let V and W be finite-dimensional F-vector spaces, and let TV-W be a linear transformation. Then 3 linear transformation S: W → V such that S.T = Iv iff T is onto (b) Let A & M mxn (F). Then 3 a matrix 13E Mnxm (F) st. BA = In iff rank(A) = n. a
Transcribed Image Text:that if I is invertible, then real с numbers there are at most h in vertible. sit. T+cs is not 6. Prove the following statements (a) Let V and W be finite-dimensional F-vector spaces, and let TV-W be a linear transformation. Then 3 linear transformation S: W → V such that S.T = Iv iff T is onto (b) Let A & M mxn (F). Then 3 a matrix 13E Mnxm (F) st. BA = In iff rank(A) = n. a
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