7. Let W = Span{x1, x2}, where x1 = [1 2 4]" and X2 – [5 5 5]" a. (4 pts) Construct an orthogonal basis {V1, V2} for W. b. (4 pts) Compute the orthogonal projection of y = [0 1]' onto W. C. (2 pts) Write a vector V3 such that {V1, V2, V3} is an orthogonal basis for R", where vi and v2 are the vectors computed in (a).
The set x1, x2} is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthonormal basis for W exactly as described in the book. Instructions: You must perform the process by using the first vector in the list as X1 and the second vector as x2. The answer is unique! Round your answer to three decimal places. 3 2 1 -1 -9 X1 X2= -6 -6 0.309 0.154 V1 V2 -0.154 -0.926 The set x1, x2}...
Let x = [xı x2 x3], and let TER → R be the linear transformation defined by T() = x1 + 6x2 – x3 -X2 X1 + 4x3 Let B be the standard basis for R2 and let B' = {V1, V2, V3}, where 7 7 and v3 = 7 V1 V2 [] --[] 0 Find the matrix of I with respect to the basis B. and then use Theorem 8.5.2 to compute the matrix of T with respect to...
Let x = [X1 X2 X3], and let T:R3 → R3 be the linear transformation defined by x1 + 5x2 – x3 T(x) - X2 x1 + 2x3 Let B be the standard basis for R3 and let B' = {V1, V2, V3}, where 4 4. ---- 4 and v3 -- 4 Find the matrix of T with respect to the basis B, and then use Theorem 8.5.2 to compute the matrix of T with respect to the basis B”....
0 5 The set of vectors {x1, x2} spans a subspace W of R3, where x1 = 19- and X 2 -- 2 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and normalize it...
3 The two vectors X1 = 0 -1 8 X2 = 5 -6 form a basis for a subspace w of Rº. Use the Gram-Schmidt process to produce an orthogonal basis for W, then normalize that basis to produce an orthonormal basis for W.
The set of vectors {x1, x2} spans a subspace W of R’, where x1 = 4 2 5 and x2 ܕ ܩ ܟ 6 -7 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and...
The set of vectors {x1, x2} spans a subspace W of R’, where x1 = 4 2 5 and x2 ܕ ܩ ܟ 6 -7 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and...
2 = -4 and x3 = 0, with p = 1 and W = span{x1, X2, X3}. 4) Let x1 = -2, X2 3 (a) W is a subspace of R". What is n? (b) Find a basis for W. (c) Isp EW? (d) Give a geometric description of W.
#8. Let W be the subspace of R3 spanned by the two linearly independent vectors v1 = (-1,2,2) and v2 = (3, -3,0). (a) Use the Gram-Schmidt orthogonalization process to find an orthonormal basis for W. (b) Use part (a) to find the matrix M of the orthogonal projection P: R W . (c) Given that im(P) = W, what is rank(M)?