201 Quiz 2 THUR solutions

# Linear Algebra with Applications (3rd Edition)

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110.201 Quiz 2 Solutions February 28, 2005 Problem 1 The first part’s answer is L = { t~v : t R } To find proj L ~w use the formula proj L ~w = 1 k ~v k 2 ( ~w · ~v ) ~v = 31 49 6 2 3 Problem 2 We first reduce A to rref: 2 1 0 1 - 1 3 - 1 0 1 ---------→ ( III ) + ( II ) * 2 1 0 0 - 1 4 - 1 0 1 ---------→ 2( III ) * + ( I ) 2 1 0 0 - 1 4 0 1 2 -----------→ - 2( III ) + ( II ) * 2 1 0 0 - 3 0 0 1 2 -----------------→ (1 / 3)( II ) + ( I ) * , ( III ) * 2 0 0 0 - 3 0 0 0 2 ----------------------→ - (1 / 3)( II ) * , (1 / 2)(( III ) * , ( I ) * 1 0 0 0 1 0 0 0 1 Since A can be rowreduced to the 3 by 3 identity matrix, it is invertible. Now perform these row operations on the three-dimensional identity matrix to obtain A - 1 : A - 1 = 1 / 6 1 / 6 - 1 / 2 2 / 3 - 1 / 3 1 1 / 6 1 / 6 1 / 2 1

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Problem 3 V is the plane orthogonal to the unit vector ~v = 1 / 6[2 , - 1 , 1]. To find a linear transformation T A : R 3
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