201 - CHAPTER 2 MATRICES AND LINEAR TRANSFORMATIONS Example...

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1 CHAPTER 2 MATRICES AND LINEAR TRANSFORMATIONS Example Multiplying A α + β to any v has the same effect as multiplying A and A to the same v in serial. In general, given A R m × n and B R n × p , wish to find C R m × p such that C v = A ( B v ) for all v R p , or simply, C = AB . (Note that such a C often does not satisfy C = BA .) Proof Suppose there is another Q R m × p such that Q v = A ( B v ) for all v R p . Then q j = Q e j = A ( B e j ) = A b j = c j , i.e., Q = C . dimension requirement: Proposition:
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2 The Row-Column Rule for the Proof You show (a), (d), (e), and (f). (b) dimension check ( AC ) P : [( k × m )( m × n )]( n × p ) = ( k × p ) A ( CP ): ( k × m )[( m × n )( n × p )] = ( k × p ); u j column j of CP = C p j column j of A ( CP ) = A ( C p j ), also, column j of ( AC ) P = ( AC ) p j = A ( C p j ) by definition. (c) you do the dimension check; column j of ( A + B ) C = ( A + B ) c j = A c j + B c j = column j of AC + BC 1
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3 (g) you do the dimension check; the ( i , j )-entry of ( AC ) T is the ( j , i )-entry of AC , which is the ( i , j )-entry of C T A T is the product of row i of C T and column j of A T , which is
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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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201 - CHAPTER 2 MATRICES AND LINEAR TRANSFORMATIONS Example...

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