# chapter 4 answer2 - SSM Elementary Linear Algebra Section...

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SSM: Elementary Linear Algebra Section 4.11 113 (c) A reflection through the yz -plane maps (1, 0, 0) to ( 1, 0, 0), (0, 1, 0) to (0, 1, 0), and (0, 0, 1) to (0, 0, 1), so the standard matrix is 100 010 001 . ⎡⎤ ⎢⎥ ⎣⎦ 5. (a) Rotation of 90 ° about the z -axis maps (1, 0, 0) to (0, 1, 0), (0, 1, 0) to ( 1, 0, 0), and (0, 0, 1) to (0, 0, 1), so the standard matrix is 01 0 . (b) Rotation of 90 ° about the x -axis maps (1, 0, 0) to (1, 0, 0), (0, 1, 0) to (0, 0, 1), and (0, 0, 1) to (0, 1, 0), so the standard matrix is 10 0 00 1 01 0 . (c) Rotation of 90 ° about the y -axis maps (1, 0, 0) to (0, 0, 1), (0, 1, 0) to (0, 1, 0), and (0, 0, 1) to (1, 0, 0), so the standard matrix is . 7. 300 0 0 , ⎡⎤ ⎡⎤ = ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ 301 3 0 , −− ⎤⎡ ⎤ = ⎥⎢ ⎥ ⎦⎣ ⎦ 0 011 1 , = 3 1 ⎤⎡⎤ = ⎥⎢⎥ ⎦⎣⎦ The image is a rectangle with vertices at (0, 0), ( 3, 0), (0, 1), and ( 3, 1). y x 9. (a) Shear by a factor of k = 4 in the y -direction has matrix 10 41 . (b) Shear by a factor of k = 2 in the x -direction has matrix 12 . 11. (a) The geometric effect is expansion by a factor of 3 in the x -direction. (b) The geometric effect is expansion by a factor of 5 in the y -direction and reflection about the x -axis. (c) The geometric effect is shearing by a factor of 4 in the x -direction. 13. (a) 11 22 00 05 ⎤⎡ = ⎥⎢ ⎦⎣ (b) 10 10 2105 25 = (c) 180 180 0 1 180 180 1 0 1001 0 cos sin sin cos °− ° °° = = 15. (a) Since 1 , = the inverse transformation is reflection about y = x . (b) Since for 0 < k < 1, 1 1 0 0 k k = and 1 1 0 0 , k k = the inverse transformation is an expansion along the same axis. (c) Since 1 = and 1 = the inverse transformation is a reflection about the same coordinate axis. (d) Since k 0, 1 0 1 kk ⎡⎤⎡ = ⎢⎥⎢ ⎣⎦⎣ and 1 , = the inverse transformation is a shear (in the opposite direction) along the same axis.

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Chapter 4: General Vector Spaces SSM: Elementary Linear Algebra 114 17. The line y = 2 x can be represented by 2 . t t ⎡⎤ ⎢⎥ ⎣⎦ (a) 13 6 7 012 02 2 tt t t t + ⎡⎤ ⎡ ⎤ ⎡⎤ == ⎢⎥ ⎢ ⎥ ⎢⎥ + ⎣⎦ ⎣ ⎦ ⎣⎦ The image is x = 7 t , y = 2 t , and using 1 7 , tx = this is the line 2 7 . yx = (b) 1 2 10 0 2 = The image is x = t , y = t which is the line y = x . (c) 01 2 10 2 ⎡⎤ ⎡⎤ = ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ The image is x = 2 t , y = t , which is the line 1 2 . = (d) 2 −− = The image is x = t , y = 2 t , which is the line y = 2 x .
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## This note was uploaded on 01/11/2012 for the course MATH 100 taught by Professor Loveys during the Fall '11 term at McGill.

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chapter 4 answer2 - SSM Elementary Linear Algebra Section...

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