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# 5.4 - 5.4 Eigenvectors and Linear Transformations Question...

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5.4 Eigenvectors and Linear Transformations Question: How do we interpret evs/e-vectors and the matrix factorization A = P DP 1 in terms of linear trans- formations? Need to build up to the answer to this big question. I. Matrix of a Linear Transformation: Consider linear transformation T : V W where subspaces V ⊆ R n and W ⊆ R m have ordered bases B = { vector b 1 , vector b 2 , · · · , vector b n } and C , respectively. T T ( vectorx ) vectorx V W R n R m [ T ( vectorx )] C [ vectorx ] B 1

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Question: How are [ vectorx ] B and [ T ( vectorx )] C related? That is, how is vectorx in terms of the B coordinates related to T ( vectorx ) in terms of the C coordinates Answer: If [ vectorx ] B = r 1 . . . r n then from superposition T ( vectorx ) = T ( r 1 vector b 1 + · · · + r n vector b n ) = r 1 T ( vector b 1 ) + · · · + r n T ( vector b n ). Using the C coordinates in W , [ T ( vectorx )] C = r 1 [ T ( vector b 1 )] C + · · · + r n [ T ( vector b n )] C , or [ T ( vectorx )] C = bracketleftBig [ T ( vector b 1 )] C [ T ( vector b 2 )] C · · · [ T ( vector b n )] C bracketrightBig r 1 . . . r n or [ T ( vectorx )] C = M [ vectorx ] B M is the matrix for T relative to the bases B and C 2
vectorx T ( vectorx ) [ vectorx ] B [ T ( vectorx )] C T multiplication by M EX: If B = { vector b 1 , vector b 2 } and C = { vector c 1 ,vector c 2 ,vector c 3 } are bases for vector spaces for V and W , respectively, and T : V W is a linear transformation with the property that T ( vector b 1 ) = 2 vector c 1 3 vector c 2 + vector c 3 , T ( vector b 2 ) = 4 vector c 1 + 5 vector c 2 + vector c 3 , find the matrix for T relative to B and C . Solution: Since [ T ( vector b 1 )] C = 2 3 1 and [ T ( vector b 2 )] C = 4 5 1 , the matrix is bracketleftBig [ T ( vector b 1 )] C [ T ( vector b 2 )] C bracketrightBig = 2 4 3 5 1 1 3

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II. Linear Transformations from V into V When W = V and C = B , M is the matrix for T relative to B , or the B - matrix for T – call it [ T ] B , so [ T ( vectorx )] B = [ T ] B [
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5.4 - 5.4 Eigenvectors and Linear Transformations Question...

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