Show that the set of vectors {ū1 = (1,1,1,1), Ū2 = (1,0,-1,0), Ūz = (0,1,0, -1)}...
LINEAR ALGEBRA
1. (4 marks) Show that the set of vectors {ū = (1,1,1,1), űz = (1,0, -1,0), vz = (0,1,0, -1)} is orthogonal. Use those vectors in the set to get an orthonormal set {w1, W2, W3}.
1. (4 marks) Show that the set of vectors {ős = (1,1,1,1), in = (1,0, -1,0), öz = (0,1,0, -1)} is orthogonal. Use those vectors in the set to get an orthonormal set {wi, wz, ws}.
1. (4 marks) Show that the set of vectors {7: = (1,1,1,1), 7x = (1,0,−1,0), 7: = (0,1,0, -1)} is orthogonal. Use those vectors in the set to get an orthonormal set {to, to, s}. 2. (6 marks) Find the best line y =c+dt to fit y=1, 1, 2, 2 at times t = -1, 0, 1, 2. (Use the least squares approximation.)
1. Show that {ū1, ū2, ū3} is an orthogonal basis for R3, and write ž as a linear combination of the vectors {ū1, ū2, ū3}, 1 ū1 -[:] -2 -2 ū2 = [ ] ū3 = Ž -11 -3 -17
-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. -
1. Determine whether the followings statements are true or false. (Com- ment: no reason needed.) (a) If the vectors ū1, ū2, üz are linearly independent, then the vectors ū1, ū2 are linearly independent as well. (b) The set {1,1 + x, (1 + x)} is a basis for P2. (c) For every linear transformation T: RM + R", there is an m xn matrix such that Tū = A✓ for all ū in R”. (d) If w1, W2 are vectors...
(1 point) Let 0.5 0.5 0.5 0.5 Ūi = 0.5 0.5 Ū2 = ū3 -0.5 0.5 2 2 -0.5 -0.5 0.5 -0.5 Find a vector ū4 in R4 such that the vectors ū1, 72, 73, and ū4 are orthonormal. 04
Determine whether the set of vectors is orthonormal. If the set is only orthogonal, normalize the vectors to produce an orthonormal set. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. O A. The set of vector is orthogonal only. The normalized vectors for u, and un U1 دادن داده هادی and uz = 0 are and respectively. 1 wa (Type exact answers, using radicals as needed.) OB. The set of vectors...
5. For parts (a)-(d) below, consider the set of vectors B = {(1,2), (2, -1)}. (a) (2 points) Demonstrate that B is an orthogonal set in the Euclidean inner product space R2. (b) (3 points) Use the set B to create an orthonormal basis in the Euclidean inner product space R2 (e) (7 points) Find the transition matrix from the standard basis S = {(1,0),(0,1)} for R2 to the basis B. Show all steps in your calculation. (d) (7 points)...
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.