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iii) There exist a unique subspace Z_3 of V such that V = W ? Z_3 and w ? z = 0, ? w ? W & z ? Z_3 i.e. Z_3 is orthogonal complement of W. Question: Is Z_3 a unique subspace? Assume W ? Z_3 = W ? Z_3' = V where both Z_3 and Z_3' are orthogonal to W. Consider z ? Z_3. Then V = W ? Z_3' ? ? w ? W and z' ? Z_3' with z = w + z' ? <z, w> = <w, w> + <z', w> ? 0 = <w, w> + 0 (by definition) ? <w, w> = 0 ? w = 0. ? z = z' ? Z_3' ? Z_3 ? Z_3'. Similarly we can show, Z_3 ? Z_3' and hence Z_3 = Z_3'. <i), ii)> are easy!!! Take V = R^2 with standard inner product. W = { (a, 0) : a ? R }. Z_1 = { (0, b) : b ? R } Z_2 = { (a, b) : a = b, a, b ? R }. Then clearly, V = W + Z_1 = W + Z_2 and V = W ? Z_1 = W ? Z_2. So, uniqueness is not guaranteed!!!

          iii) There exist a unique subspace Z_3 of V such that V = W ? Z_3 and w ? z = 0, ? w ? W & z ? Z_3 i.e. Z_3 is orthogonal complement of W. Question: Is Z_3 a unique subspace? Assume W ? Z_3 = W ? Z_3' = V where both Z_3 and Z_3' are orthogonal to W. Consider z ? Z_3. Then V = W ? Z_3' ? ? w ? W and z' ? Z_3' with z = w + z' ? <z, w> = <w, w> + <z', w> ? 0 = <w, w> + 0 (by definition) ? <w, w> = 0 ? w = 0. ? z = z' ? Z_3' ? Z_3 ? Z_3'. Similarly we can show, Z_3 ? Z_3' and hence Z_3 = Z_3'. <i), ii)> are easy!!! Take V = R^2 with standard inner product. W = { (a, 0) : a ? R }. Z_1 = { (0, b) : b ? R } Z_2 = { (a, b) : a = b, a, b ? R }. Then clearly, V = W + Z_1 = W + Z_2 and V = W ? Z_1 = W ? Z_2. So, uniqueness is not guaranteed!!!

iii) There exist a unique subspace Z3 of V such that V = W ? Z3 and w ? z = 0, ? w ? W z ? Z3 i.e. Z3 is orthogonal complement of W. Question: Is Z3 a unique subspace? Assume W ? Z3 = W ? Z3' = V where both Z3 and Z3' are orthogonal to W. Consider z ? Z3. Then V = W ? Z3' ? ? w ? W and z' ? Z3' with z = w + z' ? <z, w> = <w, w> + <z', w> ? 0 = <w, w> + 0 (by definition) ? <w, w> = 0 ? w = 0. ? z = z' ? Z3' ? Z3 ? Z3'. Similarly we can show, Z3 ? Z3' and hence Z3 = Z3'. <i), ii)> are easy!!! Take V = R^2 with standard inner product. W = (a, 0) : a ? R . Z1 = (0, b) : b ? R Z2 = (a, b) : a = b, a, b ? R . Then clearly, V = W + Z1 = W + Z2 and V = W ? Z1 = W ? Z2. So, uniqueness is not guaranteed!!!

Added by Connie H.

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