MATH 255, WINTER 2016 HOMEWORK 5 DUE FRIDAY, FEBRUARY 19 1. Let F be a eld and let V = F [x]/ x4 + 7x3 + 3x + 2 . We showed in class that as a vector space over F , V is a 4 dimensional vector space with basis 1, x, x2 , x3 . Let T : V V be the map given by multiplication by the class of 2x3 + x2 + 3. Prove that T is a linear transformation and determine the matrix corresponding to T with respect to the above basis. 2. Use the Gram-Schmidt process to nd orthonormal bases for subspaces of R4 generated by the following vectors. (a) (1, 1, 1, 0), (1, 1, 2, 1). (b) (1, 1, 0, 0), (0, 1, 1, 0), (0, 0, 1, 1). (c) (1, 1, 1, 1), (1, 1, 1, 1), (1, 1, 1, 1). 3. Let V be a nite dimensional space with an inner product, and let W be a subspace of V . Let W be the set of all vectors v V such that (v, w) = 0 for all w W . Prove that dim(W ) + dim(W ) = dim(V ). 4. We dened orthogonal transformations of a nite dimensional vector space V with an inner product using the notion of length, rather than orthogonality. Let T be a linear transformation that preserves orthogonality, i.e. if v, w V are such that (v, w) = 0 then (T v, T w) = 0. Prove that T is a scalar multiple of an orthogonal transformation. 5. Let v1 , . . . , vm be an orthonormal set of vectors in a vector space V , and let v be a vector such that v Span{v1 , . . . , vm }. Prove that the vector / n v =v (v, vi )vi i=1 given by the Gram-Schmidt process has the shortest length among all vectors of the form v x, with x Span{v1 , . . . , vm }. 1