2) Let 2 = -- 0 -- 0 -- 0 1 = W2 2 ]...

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2) Let 2 = -- 0 -- 0 -- 0 1 = W2 2 " ] 1 and let W = span{W1, W2} be the subspace of R with the standard inner product. Write the vector v as w+u with w W and u W. Find the distance between v and W. = == Solution: Let w = xW+x2W2. Then u = vw, and the condition that u W becomes (u, W) (u, w2) = 0, i.e., x1 (W, Wi) + x2 (W2, W;) = (V, W;) for i = 1,2. Computing the inner products, we obtain the linear system 2x1 + x2 = 3, x1+5x2 = 4. Solving this (e.g., by Cramer's rule), we arrive at x = 11 9 x2 = 5 so that 9' 16 2 1 Wx1W1 + x2W2 = 11 u v w = -2 " 9 9 10 The distance between v and W is, by definition, ||u|| = 1 3 2) Let 2 = -- 0 -- 0 -- 0 1 = W2 2 " ] 1 and let W = span{W1, W2} be the subspace of R with the standard inner product. Write the vector v as w+u with w W and u W. Find the distance between v and W. = == Solution: Let w = xW+x2W2. Then u = vw, and the condition that u W becomes (u, W) (u, w2) = 0, i.e., x1 (W, Wi) + x2 (W2, W;) = (V, W;) for i = 1,2. Computing the inner products, we obtain the linear system 2x1 + x2 = 3, x1+5x2 = 4. Solving this (e.g., by Cramer's rule), we arrive at x = 11 9 x2 = 5 so that 9' 16 2 1 Wx1W1 + x2W2 = 11 u v w = -2 " 9 9 10 The distance between v and W is, by definition, ||u|| = 1 3