Topic 13C-lin-trans-3.pdf - Topic 13C Linear Transformation...

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Topic 13C Linear Transformation III Composition and Invertibility In this topic, we will consider the composition of linear transformations and show how this links nicely with matrix multiplication. We will define the inverse of a matrix and relate this to the invertibility of the function determined by the matrix. Definition 8: Composition of functions Let T 1 : F n F m and T 2 : F m F p be functions. We define the function T = T 2 T 1 , T : F n F p , by T ( x ) = ( T 2 T 1 )( x ) = T 2 ( T 1 ( x )) . T is called the composite function of T 2 and T 1 . Lemma 9: Composition of linear transformation is linear. Let T 1 : F n F m and T 2 : F m F p . If T 1 and T 2 are both linear transformations, then the composite function of T 2 and T 1 is also a linear transformation. Proof Since T 1 is linear, we have: T 1 ( c 1 x 1 + c 2 x 2 ) = c 1 T 1 ( x 1 ) + c 2 T 1 ( x 2 ) , for all x 1 , x 2 F n , c 1 , c 2 F . Since T 2 is linear we have: T 2 ( d 1 y 1 + d 2 y 2 ) = d 1 T 2 ( y 1 ) + d 2 T 2 ( y 2 ) , for all y 1 , y 2 F m , d 1 , d 2 F . Consider ( T 2 T 1 )( c 1 x 1 + c 2 x 2 ) = T 2 ( c 1 T 1 ( x 1 ) + c 2 T 1 ( x 2 )). We can let T 1 ( x 1 ) = y 1 , and T 1 ( x 2 ) = y 2 , so that: ( T 2 T 1 )( c 1 x 1 + c 2 x 2 ) = T 2 ( c 1 y 1 + c 2 y 2 ) = c 1 T 2 ( y 1 ) + c 2 T 2 ( y 2 ) , by linearity of T 2 = c 1 T 2 ( T 1 ( x 1 )) + c 2 T 2 ( T 1 ( x 2 )) and we conclude that ( T 2 T 1 )( c 1 x 1 + c 2 x 2 ) = c 1 ( T 2 T 1 )( x 1 ) + c 2 ( T 2 T 1 )( x 2 ) , i.e. ( T 2 T 1 ) is linear. 1
Lemma 10: The matrix of the composite function. Let T 1 : F n F m and T 2 : F m F p be linear transformations. Let the composite function of T 2 and T 1 be T , that is, let T = T 2 T 1 . Then [ T ] S = [ T 2 T 1 ] S = [ T 2 ] S [ T 1 ] S . Proof Let the standard basis for F n be { e 1 , e 2 , . . . , e n } and the one for F m be { f 1 , f 2 , . . . , f m } . Suppose that we let: A = [ T 1 ] S , so that a kj = ( T 1 ( e j )) k and B = [ T 2 ] S , so that b ik = ( T 2 ( f k )) i . In order to obtain [ T ] S , we must apply T to each of the vectors in { e 1 , e 2 , . . . , e n } , we then have: ([ T ] S ) ij = ( T ( e j )) i by definition of the matrix representation = (( T 2 T 1 )( e j )) i = T 2 m X k =1 a kj f k !! i by definition of a kj = m X k =1 a kj T 2 ( f k ) ! i using the linearity of T 2 = m X k =1 a kj ( T 2 ( f k )) i using the linearity of taking a component = m X k =1 a kj b ik = m X k =1 b ik a kj = ( BA ) ij by the definition of matrix multiplication = ([ T 2 ] S [ T 1 ] S ) ij . Since all their entries are equal for all i = 1 , . . . , p and j = 1 , . . . , n , we conclude that the two matrices are identical, and thus [ T ] S = [ T 2 ] S [ T 1 ] S . This gives us a very efficient way of obtaining the matrix representation of a composite function. It also explains the reasons behind the definition of matrix multiplication. 2
Example 15 Let A = 1 + i - 3 i - 2 2 1 + 2 i 4 i be a matrix in M 2 × 3 ( C ) and B = 3 i 1 + i 1 0 1 - i 3 - 2 i be a matrix in M 3 × 2 ( C ). Let T A and T B be the functions determined by A and B , respectively. What is the standard matrix of their composite function, T B T A ?

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