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e506fk - V . Include the denition of L-1 ( T ) in your...

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MAS 3105 Nov 8, 2006 Quiz 5 Key Prof. S. Hudson 1) Answer TRUE or FALSE: As usual, R 2 E means column vectors in R 2 , written using the basis E . If L : R 1 R 1 is linear, then L ( x ) = L (1) x , for all x R 1 . The transformation L ( A ) = A T is linear, from R 3 x 2 to R 2 x 3 . If A is similar to I , then A = I ( I is the identity matrix). If A represents L : R 2 E R 2 E and B represents L : R 2 F R 2 F , then A is similar to B . If A represents L : R 2 E R 2 F and B represents L : R 2 F R 2 E , then A is similar to B . 2) Find the matrix representation of L : R 2 R 2 , where L is a 90 degree clockwise rotation [use the standard basis of R 2 ]. 3) Choose ONE of these. You can answer on the back. a) Prove thm 5.1.1; that in R 2 , x T y = || x || || y || cos( θ ). Include a picture. b) HW 4.2.20: Assuming that L : V W is linear and T is a subspace W , prove that L - 1 ( T ) is a subspace of
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Unformatted text preview: V . Include the denition of L-1 ( T ) in your answer (this will also justify some partial credit). Remarks and Answers: The average was about 40/60. As = 48 to 60, and subtract about 6 points per letter. I wrote your estimated semester grade in the corner, based on your best 4 out of 5 quiz scores, so far. 1) TTTTF 2) Similar to Ch 4.2.5. Compute L ( e 1 ) = [0 ,-1] T to get the rst column, etc: A = 1-1 3a) See the textbook. For full credit, explain every step. Mention the Law of Cosines by name, and that || x || 2 = x T x and x T y = y T x . 1...
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