Lecture 23 - Mar 4

Lecture 23 - Mar 4 - Wednesday March 4 Lecture 23 : The...

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Wednesday March 4 Lecture 23 : The inverse of a linear transformation. (Refers to section 3.5) Expectations: 1. Recognize that an invertible matrix has RREF equal to I . 2. Recognize that if A is the matrix induced by T and A is invertible then T is invertible and T -1 induces the matrix A -1 . 3. Use a row-reduction algorithm to find the inverse of a matrix A . 23.0 Propostion – If T is a 1 – 1 linear transformation mapping R n into R n then T -1 is linear. Proof: Suppose y 1 and y 2 belong to the image of T . such that T ( x 1 ) = y 1 and T ( x 2 ) = y 2 . Then T -1 ( a y 1 + b y 2 ) = T -1 ( a T ( x 1 ) + b T ( x 2 )) = T -1 ( T ( a x 1 + b x 2 )) = a x 1 + b x 2 = a T -1 ( y 1 ) + bT -1 ( y 2 ) So T -1 is linear. 23.1 Theorem Let T : R n R n be a linear transformation. Let A be the standard matrix induced by T . Then T has an inverse T -1 iff the matrix A is invertible. Furthermore, the matrix induced by T -1 is A -1 . Proof : ( ) Suppose T is a linear transformation and T -1 is its well defined inverse. Let A be the standard matrix induced by T ; let B be the standard matrix induced by T -1 .
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Then for all x in R n , x = T -1 ( T ( x )) = B ( A ( x )) and x = T ( T -1 ( x )) = A ( B ( x )) . Thus AB ( x ) = BA ( x ) = I ( x ) for all x and so AB = BA = I . and so B = A -1 .
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Lecture 23 - Mar 4 - Wednesday March 4 Lecture 23 : The...

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