2114-Homework4.pdf

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Math 2114 Inverses Homework 4 Name: 1. Find 1 2 0 0 - 1 2 2 3 1 - 1 by hand. Show all steps. 2. Let B = 1 3 0 0 - 1 2 and define S : R 2 R 2 by S ( ~x ) = B~x . Find the images under S of ~x = (9 , 2) and ~ y = (0 , - 3) and the vector ~x + ~ y . Also sketch the three vectors before and after the transformation. Write 1–2 sentences that explain how this problem relates to the first problem from the last homework. 3. Assume that T is a linear transformation. Remember that ~e 1 = (1 , 0) and ~e 2 = (0 , 1). For each part, find the standard matrix B for T - 1 , draw the image of the “half-shaded unit square (shown below) under T - 1 , and explain the effect of T - 1 in words. x 1 x 2 T ? (a) T : R 2 R 2 rotates points (about the origin) through - π/ 4 radians (this is a clockwise rotation). (b) T : R 2 R 2 maps ~e 1 onto ~e 1 - ~e 2 but leaves ~e 2 unchanged. (c) T : R 2 R 2 first reflects points across the vertical axis and then rotates points π/ 2 radians (counterclockwise). 4. TRUE OR FALSE: If the columns of A are linearly dependent, then so are the rows of
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