06-inverse

06-inverse - 10/8/2003, THE INVERSE Math 21b, O. Knill...

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Unformatted text preview: 10/8/2003, THE INVERSE Math 21b, O. Knill HOMEWORK: Section 2.3: 10,20,30,40*,42*, Section 2.4: 14,28,40*,72* INVERTIBLE TRANSFORMATIONS. A map T from X to Y is invertible if there is for every y Y a unique point x X such that T ( x ) = y . EXAMPLES. 1) T ( x ) = x 3 is invertible from X = R to X = Y . 2) T ( x ) = x 2 is not invertible from X = R to X = Y . 3) T ( x, y ) = ( x 2 + 3 x- y, x ) is invertible from X = R 2 to Y = R 2 . 4) T ( ~x ) = Ax linear and rref( A ) has an empty row, then T is not invertible. 5) If T ( ~x ) = Ax is linear and rref( A ) = 1 n , then T is invertible. INVERSE OF LINEAR TRANSFORMATION. If A is a n n matrix and T : ~x 7 Ax has an inverse S , then S is linear. The matrix A- 1 belonging to S = T- 1 is called the inverse matrix of A . First proof: check that S is linear using the characterization S ( ~a + ~ b ) = S ( ~a ) + S ( ~ b ) , S ( ~a ) = S ( ~a ) of linearity. Second proof: construct the inverse using Gauss-Jordan elimination....
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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