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Unformatted text preview: )cos(2 ) sin(2 ) cos(2 ) cos(2 ) + sin(2 ) sin(2 ) = cos (2(  ))sin (2(  )) sin (2(  )) cos (2(  )) . Thus R R is a rotation by 2(  ). Similarly, R R is a rotation by 2(  ). These rotations are equal when and are congruent modulo / 2. 2. [4 marks] Let K : R m R n and L : R n R m be linear transformations, and let m < n . Prove that the reduced rowechelon form of the standard matrix of K L is not an identity matrix. Let K and L have standard matrices A and B , respectively. AB , the standard matrix of K L , is m m . Now the reduced rowechelon form of AB is the m m identity matrix if, and only if, Nul( AB ) = { } . Since m < n , B has at least one free variable. Hence there exists a nonzero vector x R m such that B x = , and AB x = . Thus Nul( AB ) 6 = { } ....
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This note was uploaded on 03/19/2009 for the course M m136 taught by Professor Fokshuen during the Spring '09 term at Waterloo.
 Spring '09
 FokShuen
 Algebra, matlab, Addition

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