704 - Definition Let V be a finite-dimensional vector space...

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1 Property: [ u + v ] B = [ u ] B + [ v ] B and [ c u ] B = c [ u ] B for all u , v V and scalar c . Definition Let V be a finite-dimensional vector space and B be a basis for V . For any vector v in V , the vector Φ B ( v ) is called the coordinate vector of v relative to B and is denoted as [ v ] B . Let T : V V be a linear operator on an n -dimensional vector space V with a basis B . Define the linear operator Φ B T ( Φ B ) 1 : R n R n , and consider its standard matrix A , called the matrix representation of T with respect to B and denoted as [ T ] B . With the notations, [ T ] B = A and T A = Φ B T ( Φ B ) 1 . V V R n R n Φ B ( Φ B ) 1 T Φ B T ( Φ B ) 1 Example: Let T : P 2 P 2 be defined by T ( p ( x )) = p (0) + 3 p (1) x + p (2) x 2 for all p ( x ) in P 2 . Then T is linear. For B = {1, x , x 2 }, [ T ] B = A = [ a 1 a 2 a 3 ] and 22 12 31 2 3 10 [( 1 ) ] [ 13 ] 3, [() 3 2 ] 01 0 0 [ ( )] [3 4 ] 3 , so [ ] [ ] 3 3 3 . 41 2 4 Tx x T x x x x x T ⎡⎤ ⎢⎥ == + + = = = + = ⎣⎦ ⎤⎡ ⎥⎢ + = = = ⎦⎣ aa a a BB B B B Property: If B = { v 1 , v 2 , , v n } , then [ T ] B = [ [ T ( v 1 )] B [ T ( v 2 )] B " [ T ( v n )] B ]. Proof [ T ] B = A A e j = T A ( e j ) = Φ B T ( Φ B ) 1 ( e j ) = Φ B T ( v j ) = [ T ( v j )] B .
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2 Proof [ T ( v )] B = Φ B T ( v ) = Φ B T ( Φ B ) 1 Φ B ( v ) = T A ([ v ] B ) = [ T ] B [ v ] B , where A = [ T ] B . 10 Example: Let V = Span B , where B = { e t cos t , e t sin t } is L.I. and thus a basis of V , and the linear operator D : V V be defined by D ( f ) = f for all f V . Then = + = + = 1 1 1
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This note was uploaded on 10/16/2010 for the course EE 155 taught by Professor Fong during the Spring '09 term at National Taiwan University.

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704 - Definition Let V be a finite-dimensional vector space...

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