Section 4 Notes - a b c Ex 2.3.35 Consider the upper...

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MATH 294 Chapter 2.3 The Inverse of a Linear Transformation Ex 2.3.35 : Consider the upper triangular 3 × 3 matrix A = a b c 0 d e 0 0 f . For which choices of a, b, c, d, e, and f is A invertible? Is the inverse matrix upper triangular? Solution: To answer the question we use Fact 2.3.5. Let us form the matrix B = [ A . . . I 3 ] and compute rref(B). a b c . . . 1 0 0 0 d e . . . 0 1 0 0 0 f . . . 0 0 1 ÷ a ÷ d ÷ f -→ 1 b a c a . . . 1 a 0 0 0 1 e d . . . 0 1 d 0 0 0 1 . . . 0 0 1 f - e d 3 rd -→ 1 b a c a . . . 1 a 0 0 0 1 0 . . . 0 1 d - e fd 0 0 1 . . . 0 0 1 f - b a 2 nd - c a 3 rd -→ 1 0 0 . . . 1 a - b da eb fda - c af 0 1 0 . . . 0 1 d - e fd 0 0 1 . . . 0 0 1 f = = [ I 3 . . . A - 1 ]. Therefore, matrix A is invertible if and only if diagonal elements a, d, f are different from 0 (because we had to divide by these constants). We found that the inverse of A is an upper triangular matrix too. Using the same reasoning ( step1 : divide every row by a corresponding diagonal element; step2: get the leading ones by subtracting corresponding rows ) it’s pretty easy to figure out that an upper triangular matrix of arbitrary size is invertible if and only if its diagonal elements do not equal zero. Think of what are the differences in case of a lower triangular matrix at home. ================================================================= Ex 2.3.41(modified) Which of the following linear transformations T from R 2 to R 2 are invertible? Find the inverse if it exists. a. Reflection in a line. Invertible (see the graph). b. Projection onto a line. Non-invertible (see the graph). c. Scaling by 5. y = T ( x ) = 5 x T - 1 ( y ) = y 5 . Hence, it’s invertible. d. Rotation about an axis. Invertible (see the graph). Think of the same transformations in case of a mapping from R 3 to R 3 .
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