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-12 -4 8. (4 pts) Let ū1 = -12 and U2 = -15 These vectors form a basis for a subspace -6 -11 V of R3. Starting with 71, 72, use the Gram-Schmidt process to find an orthonormal basis ū1, ū2 for V. -
) Let A be the following matrix: 13 0 2 0 2 2 0 0 6 (a) Enter its characteristic equation below. Note you must use p as the parameter instead of , and you must enter your answer as a equation, with the equals sign. (b) Enter the eigenvalues of the matrix, including any repetition. For example 16,16,24. 5 (c) Find the eigenvectors, and then use Gram-Schmidt to find an orthonormal basis for each eigenvalue's eigenspace. Build an orthogonal...
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1 (10 points) Show that {u1, U2, U3} is an orthogonal basis for R3. Then express x as a linear 3 4 combination of the u's. u -3 U2 = 0 ,u3 5 6 -2 2 -1 (10 points) Suppose a vector y is orthogonal to vectors u and v. Prove that y is orthogonal to the vector 4u - 3v. 10. (2 points each) True or False: ( ) Eigenvalues must be nonzero scalars. ( )...
1 -1.2 5 Uį = U2 = -3 1, U3 = 2 , 14 = 29 ( 7 Answer the following questions and give proper explanations. (a) Is {ui, U2, uz} a basis for R3? (b) Is {ui, U2, u4} a basis for R4? (c) Is {ui, U2, U3, U4, u; } a basis for R? (d) Is {ui, U2, U3, u} a basis for Rº?! (e) Are ui, u, and O linearly independent?! Problem 6. (15 points). Let A...
[0 1 1 1 0 1 1 1 1 0 1 1. Let A and b 1 1 1 0 Find the eigenvalues of A. Find four independent eigenvectors of A. Ar for each eigenvector T. Verify that Ar Find the coordinates of b in the eigenbasis. . Find the matrix of A relative to the eigenbasis. Find a matrix P such that PAP is diagonal. Find four ort hogonal eigenvectors of A Find the coordinates of b in the...
9. (2 pts per part) Let A be an m x n matrix, where m > n, and suppose that the rank of A is n (i.e., A has full column rank). Briefly justify your answers to each question below. a. Which two of the following statements are true? i. There are no vectors in Nul(A). ii. There is no basis for Nul(A). iii. dim(Nul(A)) = 0 iv. dim(Nul(A)) = m – n b. Are the columns of A a...
2 1 3 4 -2 5 7 -2 9 Problem 9 Let uj = u2 = 13 2 Also let v= 0 5 3 10 -6 0 11 1 1 7 a) (4 pts) Compute prw(v) where W = Span{u1, U2, U3} CR5. b) [4 pts) Compute prw(v) where w+ denotes the orthogonal complement of W in R5. c) [3 pts) Compute the distance between v and W.
9 -4 0 0 A4 5 2 0 0 0 1 2 and consider the vector space R4 with the inner product given by v, w)Aw. Let 0 0 -2 and let W span(Vi, V2, V3 ). In this problem, you will apply the Gram-Schmidt procedure to vi, v2, v3 to find an orthogonal basis (u, u2, u31 for W (with respect to the above inner product). b) Compute the following inner products. (v2, u1) - Then u2 =Y2__v2.ul) ui,...
Problem 6 (18 pts.): Let A be a 4 x 2 matrix given by: -1 -5 1 1 1 A= -1 -1 1 5 a) Compute the Gram-Schmidt QR factorization of A. b) Use the QR factorization to find the least squares solution of Az = 6, where 6= (-2,-1,5,0).
6. Let P be the subspace in R 3 defined by the plane x − 2y + z
= 0. (a) [5 points] Use the Gram–Schmidt process to find orthogonal
vectors that form a basis for P. (b) [5 points] Find the projection
p of b = (3, −6, 9) onto P.
6. Let P be the subspace in R3 defined by the plan 2y+z0 (a) [5 points] Use the Gram-Schmidt process to find orthogonal vectors that form a basis...