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Math 54 - Quiz 06 Ans, Spring 2006

# Math 54 - Quiz 06 Ans, Spring 2006 - p 2 = c 2-h c 2 p 1 i...

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Math 54, Quiz 6 Solutions Sections 202/203 GSI Carter 1. For each of the following statements, write the word “true” or “false.” (a) Let P be the matrix corresponding to some orthogonal projection. Then P 5 = P 3 . True. Since P is an orthogonal projection, P 2 = P . Therefore P 5 = ( P 2 )( P 2 )( P ) = ( P )( P )( P ) = P 3 . (b) Let A be a matrix with orthogonal columns. Then A T A = I . False. The matrix 2 I has orthogonal columns, but (2 I ) T (2 I ) = 4 I = I . 2. Let c 1 = (1 , 1 , 1) and c 2 = (2 , 0 , 1). Find the matrix for the orthogonal projection onto the subspace of R 3 spanned by c 1 and c 2 . First, we need to give an orthogonal basis for the subspace spanned by c 1 and c 2 . To do this, use the Gram-Schmidt process. Let
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Unformatted text preview: p 2 = c 2-h c 2 , p 1 i h p 1 , p 1 i p 1 = (2 , , 1)-3 3 (1 , 1 , 1) = (1 ,-1 , 0) . Then for any v ∈ R 3 , the orthogonal projection is given by P ( v ) = h v , (1 , 1 , 1) i 3 (1 , 1 , 1) + h v , (1 ,-1 , 0) i 2 (1 ,-1 , 0) . Therefore, P (1 , , 0) = 1 3 (1 , 1 , 1) + 1 2 (1 ,-1 , 0) = ± 5 6 ,-1 6 , 1 3 ² P (0 , 1 , 0) = 1 3 (1 , 1 , 1)-1 2 (1 ,-1 , 0) = ±-1 6 , 5 6 , 1 3 ² P (0 , , 1) = 1 3 (1 , 1 , 1) = ± 1 3 , 1 3 , 1 3 ² . Therefore, the matrix corresponding to the orthogonal projection is 1 6   5-1 2-1 5 2 2 2 2   . 1...
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