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Therefore the dimension of v is 4 1 3 a basis b for v

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Unformatted text preview: 1 1 as ￿ = y4 y 0 . 1 1 −1 1 . A basis B for V ⊥ is a vector ￿ , a Answer: V ⊥ is the line spanned by a vector ￿ = a 0 1 that is, B = {￿ }. The dimension of V ⊥ is 1. a (b) The dimension of V is “the dimension of the large space, R4 , minus the dimension of its perpendicular complement V ⊥ .” Therefore, the dimension of V is = 4 − 1 = 3. A basis B for V can be formed either by columns of A, or by rows of any reduced form of AT . Thus, it be given, B = {￿1 , ￿2 , ￿3 } as v v v can 0 0 1 0 1 0 , . or as B = , 0 1 0 0 −1 1 Both versions of B span space V (the first by definition, second by the property of matrix reduction). They must be independent, since the dimension of V is 3. (This can be deduced either as above, or by counting number of pivots in the reduced version of the matrix AT .) (c) The formula for the matrix representing the orthogonal projection onto the column space of a matrix A is A(AT A)−1 AT . We will use it first to find P ⊥ , for which the matrix A is given by a single column ￿ . So, AT = (−1, 1, 0, 1) · (−1, 1, 0, 1) = 1 + 1 + 0 + 1 =...
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This document was uploaded on 02/19/2014 for the course MATH 441 at WVU.

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