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Suppose V is a vector space with a non-trivial subspace W, i.e., W is not {0} and W is not V. If p : V → V is a linear map given by Sv if veW 9(v) = l0 if veV\W, then show that 1- v(p(v)) = v(v) for v e V. 2- dim(Im(»)) = dim W.

Suppose V is a vector space with a non-trivial subspace W, i.e., W is not {0} and W is not V. If p : V → V is a linear map given by Sv if veW 9(v) = l0 if veV\W, then show that 1- v(p(v)) = v(v) for v e V. 2- dim(Im(»)) = dim W.

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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USING METHODS OF LINEAR ALGEBRA Suppose V is a vector space with a non-trivial subspace W, i.e., W is not {0} and W is not V. If o : V →V is a linear map given by Sv if ve W p(v) = lo if veV\W, then show that 1- v(p(v)) = v(v) for v e V. 2- dim(Im(4)) = dim W. Suppose A E Mat3x4(R) and B e Mat4xa(R) are such that rank(B) = 2, BA is the 4 x 4 zero matrix and AB is not the 3 x 3 zero matrix. Determine rank(A). Justify your answer.
Transcribed Image Text:USING METHODS OF LINEAR ALGEBRA Suppose V is a vector space with a non-trivial subspace W, i.e., W is not {0} and W is not V. If o : V →V is a linear map given by Sv if ve W p(v) = lo if veV\W, then show that 1- v(p(v)) = v(v) for v e V. 2- dim(Im(4)) = dim W. Suppose A E Mat3x4(R) and B e Mat4xa(R) are such that rank(B) = 2, BA is the 4 x 4 zero matrix and AB is not the 3 x 3 zero matrix. Determine rank(A). Justify your answer.
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