Lecturenotes23-25March2011

# Lecturenotes23-25March2011 - Lets see some more examples of...

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Let’s see some more examples of ﬁnding standard matrix of a matrix transformation Example: Find the standard matrix of the given operators 1. T : R 3 -→ R 3 , reﬂection through the xy-plane 2. T : R 3 -→ R 3 , reﬂection through the plane x=z 3. T : R 3 -→ R 3 , Dilation with factor k=2 Solution: 1. . [ T ] = 1 0 0 0 1 0 0 0 - 1 2. . n = (1 , 0 , - 1) ,T ( u ) = u - 2 proj n u = ( u 1 ,u 2 ,u 3 ) - ( u 1 - u 3 , 0 ,u 3 - u 1 ) = ( u 3 ,u 2 ,u 1 ) [ T ] = 0 0 1 0 1 0 1 0 0 1

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3. . [ T ] = k 0 0 0 k 0 0 0 k Please look at book for more examples on standard matrix of contraction, rotation, re- ﬂection, compression operators. Section 4.10 Properties of Matrix Transformations Composition of Matrix Transformations: Let T 1 : R n R k and T 2 : R k R m be two matrix transformations for which codomain of T 1 = domain of T 2 . Then for each x R n we can compute T 2 ( T 1 ( x )) : R n R m . Application of T 1 followed by T 2 produces a transformation from R n to R m . This is called the composition of T 2 with T 1 and is denoted by T 2 T 1 and T 2 T 1 ( x ) = T 2 ( T 1 ( x )) .
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Lecturenotes23-25March2011 - Lets see some more examples of...

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