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asst6 - W(e For all vectors y from part(c find the...

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MATH 235, Fall 2007 Assignment # 6 1. (a) Prove that B = 1 2 1 , 1 - 1 1 , 1 0 - 1 is an orthogonal basis for C 3 . (b) Write vectors x = 1 2 3 and y = 1 i 0 , as linear combinations of vectors from B . 2. Let W = Span { u 1 , u 2 } , where u 1 = 1 2 2 , u 2 = 2 1 - 2 . (a) Prove that { u 1 , u 2 } is an orthogonal basis for W . (b) Find the matrix form of the orthogonal projection proj W (i.e. the matrix A C 3 × 3 such that proj W ( x ) = A x for all x C 3 ). (c) For all choices of y from the set 1 2 2 , 1 1 2 , i 0 0 find y W and z W such that y = y + z . (d) For all vectors y from part (c) find the best approximations by vectors from
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Unformatted text preview: W . (e) For all vectors y from part (c) find the distance from y to W . 3. If columns of matrices U ∈ C m × n and V ∈ C n × m form orthonormal sets, then prove that columns of the matrix UV also form an orthonormal set. 1 4. If W = Span { u 1 , u 2 } , where u 1 = 1-1 and u 2 = 2 1 2 , then find all vectors y ∈ R 3 such that y · u 1 = 6, y · u 2 = 9, and dist ( y , W ) = √ 6. 2...
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