5) (20 points) a) Show that the vectors x1 = (1, 1, 0)T , x2 = (1, 0, 1)T , x3 = (1, 0, 0)T are linearly independent. Do they form a basis of R3 ? Explain.
b) Find an orthonormal basis of R3 using x1 = (1, 1, 0)T , x2 = (1, 0, 1)T and x3 = (1, 0, 0)T .
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...
1. Consider a set of vectors S = {X1, X2, X3 } in which X1 = (1,0,0), X2 = (a, 1, -a), X3 = (1, 2, 3a +1) Determine the value (or values) of a for which the set Sabove is linearly independent (LI). 2. Consider a set of vectors T = {y1, y2.ya} in which yı = (1,2,0), y2 = (1, m,5), and y3 = (0,4, n) Determine a condition on m and n such that the set T...
Question 1: Let T: R3 ---> R2 defined by T(x1,x2,x3) = (x1 + 2x2, 2x1 - x2). Show that T as defined above is a Liner Transformation. Question 2: Determine whether the given set of vectors is a basis for S = {(1,2,1) , (3,-1,2),(1,1,-1)} R3 Need answers to both questions.
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...
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.
Please show work Consider the following linear transformation T: RS → R3 where T(X1, X2, X3, X4, Xs) = (x1-X3+X4, 2x1+x2-X3+2x4, -2x1+3x3-3x4+xs) (a) Determine the standard matrix representation A of T(x). (b) Find a basis for the kernel of T(x). (c) Find a basis for the range of T(x). (d) Is T(x) one-to-one? Is T(x) onto? Explain. (e) Is T(x) invertible? Explain
The vectors X1 = = (1, -1 -1,1,-1) and X2 = = (1,1, -1, -1)7 form an orthonormal set in R4. Let 1 -1 1 -1 A= 1 -1 -1 (a) Extend the set {x1, x2} to an orthonormal basis for R4 by finding an orthonormal basis for the null space N(A) of A. (b) Give the QR-decomposition for AT.
The vectors X1 = = (1, -1 -1,1,-1) and X2 = = (1,1, -1, -1)7 form an orthonormal set in R4. Let 1 -1 1 -1 A= 1 -1 -1 (a) Extend the set {x1, x2} to an orthonormal basis for R4 by finding an orthonormal basis for the null space N(A) of A. (b) Give the QR-decomposition for AT.
5.4.3. Consider the following set of three vectors. X2? 0 (a) Using the standard inner product in 4, verify that these vec tors are mutually orthogonal (b) Find a nonzero vector x4 such that (x1, x2, x3, x4) is a set of mutually orthogonal vectors. c) Convert the resulting set into an orthonormal basis for .