
Instructions: 2 3 -4 3 Find V Let V be the plane spanned by the vectors...
6. (20 points) Let W be a plane spanned by the vectors ői = [1, 2, 2)", T2 = (-1,1,2) (a). Find an orthonormal basis for W. (b). Extend it to an orthonormal basis of R3.
4 Let u-4 , v--3, w1For what value of h is w in the plane spanned by u and v?
Let W be the subspace spanned by the given vectors. Find a basis for Wt, 0 1 A. W1 = W2 = 3 2 -1 2 B. W W2 2 -3 W3 = 6
(3 points) Let W be the subspace of R spanned by the vectors 1and 5 Find the matrix A of the orthogonal projection onto W A-
(3 points) Let W be the subspace of R spanned by the vectors 1and 5 Find the matrix A of the orthogonal projection onto W A-
Let W be the subspace of R4 spanned by the orthogonal vectors 1 0 0 ui , ua : 0 1 Find the orthogonal decomposition of v = ܝܬ ܥ 5 -4 6 with respect to W. -5 p= projw (v) = q= perpw («) =
Let E be the plane in R3 spanned by the orthogonal vectors v1=(121)and v2=(−11−1) The reflection across E is the linear transformation R:R3→R3 defined by the formula R(x) = 2 projE(x)−x (a) Compute R(x) for x=(1260) (b) Find the eigenspace of R corresponding to the eigenvalue 1. That is, find the set of all vectors x for which R(x) =x. Justify your answer.
(1 point) Find the minimal distance from the point (-6,9, 8) to the plane V of R3 spanned by-3 and 2 Give your answer correct to at least five significant figures The minimal distance is
(1 point) Find the minimal distance from the point (-6,9, 8) to the plane V of R3 spanned by-3 and 2 Give your answer correct to at least five significant figures The minimal distance is
(5 points) Let 5 -4 v= 1-3 -3 and let W the subspace of R4 spanned by ū and 7. Find a basis of W?, the orthogonal complement of W in R4.
Q6. Let W be the subspace of R' spanned by the vectors u. = 3(1, -1,1,1), uz = 5(–1,1,1,1). (a) Check that {uj,uz) is an orthonormal set using the dot product on R. (Hence it forms an orthonormal basis for W.) (b) Let w = (-1,1,5,5) EW. Using the formula in the box above, express was a linear combination of u and u. (c) Let v = (-1,1,3,5) = R'. Find the orthogonal projection of v onto W.
Find the best approximation to z by vectors of the form C7 V + c2V2. 3 1 3 -1 -6 1 z = V2 4 0 -3 3 1 The best approximation to z is . (Simplify your answer.) - 15 - 8 8 - 1 Let y = , and v2 Find the distance from y to the subspace W of R* spanned by V, and vą, given 1 0 1 - 15 3 3 - 13 09 that...