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Unformatted text preview: ECEN 601: Assignment 6 Problems: 1. Let W 1 and W 2 be subspaces of a vector space V such that the set-theoretic union of W 1 and W 2 is also a subspace. Prove that one of the spaces W i is contained in the other. 2. Let V be a vector space over a subfield F of the complex numbers. Suppose that α , β , and γ are linearly independent vectors in V . Prove that ( α + β ), ( β + γ ), and ( γ + α ) are linearly independent. 3. Show that the formula (Bigg summationdisplay j a j x j vextendsingle vextendsingle summationdisplay k b k x k )Bigg = summationdisplay j,k a j b k j + k + 1 (1) defines an inner product on the space R [ x ] of polynomials over the field R . Let W be the subspace of polynomials of degree less than or equal to n . Restrict the above inner product to W , and find the matrix of this inner product on W , relative to the ordered basis 1 ,x,x 2 ,...,x n . ( Hint: To show that the formula defines an inner product, observe that ( f | g ) = integraldisplay 1 f ( t ) g (...
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- Fall '08
- Linear Algebra, inner product, MGS