Question 1 (10 points) Projection matrix and Normal equation: Consider the vectors v1 = (1, 2,...
15 points) Consider the following vectors in R3 0 0 2 V1 = 1 ; V2 = 3 ; V3 = 1] ; V4 = -1;V5 = 4 1 2 3 = a) Are V1, V2, V3, V4, V5 linearly independent? Explain. b) Let H (V1, V2, V3, V4, V5) be a 3 x 5 matrix, find (i) a basis of N(H) (ii) a basis of R(H) (iii) a basis of C(H) (iv) the rank of H (v) the nullity...
45 points) Consider the following vectors in R3 2 0 0 2 2 Vi = 1 ;02 31; V3 = 11:04 = -1 ; Us = 4 2 2 3 (c) Find a basis of R3 among V1, V2, V3, V4, V5, and call it basis V. (d) Is vs Espan{V1, V2, 03, 04}? Explain. (e) Find the coordinates of us with respect to the basis V.
1 4 3 13 The vectors V1 = | 2 and V2 = 5 span a subspace V of the indicated Euclidean space. Find a basis for the orthogonal complement vt of V. 8 36 4 13 Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. O A. A basis for the orthogonal complement vt is {}. (Use a comma to separate vectors as needed.) OB. There is no basis for the orthogonal...
Please show work Problem 2. Consider the vectors [1] 1 1 v1 = 1, V2 = -1, V3 = -3 , 04 = , 05 = 6 Let S CR5 be defined by S = span(V1, V2, V3, V4, 05). A. Find a basis for S. What is the dimension of S? B. For each of the vectors V1, V2, V3, V4.05 which is not in the basis, express that vector as linear combination of the basis vectors. C. Consider...
Will rate once all is completed. 1) 2) 3) 4) (12 points) Find a basis of the subspace of R that consists of all vectors perpendicular to both El- 1 1 0 and 7 Basis: , then you would enter [1,2,3],[1,1,1] into the answer To enter a basis into WeBWork, place the entries. each vector inside of brackets, and enter a list these vectors, separated by commas. For instance if vour basis is 31 2 and u (12 points) Let...
Question 1 2 pts Describe the span of {(1,0,0),(0,0,1)} in R3 The x-z plane R3 R2 The x-y plane Question 2 2 pts Describe the span of {(1,1,1),(-1,-1, -1), (2,2, 2)} in R3 A plane passing through the origin Aline passing through the origin R3 A plane not passing through the origin A line not passing through the origin Question 3 2 pts Let u and v be vectors in R™ Then U-v=v.u True False Question 4 2 pts Ifu.v...
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...
1. (10 points) Consider the vectors u = 0 and v = | 2 [E (a) Find cosine of the angle between two vectors. Is the angle acute, obtuse, or neither? (b) Find p = projspan{v}u and verify that u-p is orthogonal to v.
= ) 1 5 # 4 [10 points] Recall the real symmetric 2 by 2 matrix B of # 3. 5 1 # 4 a) From the matrix fact sheet, what are the eigenvalues of B and its characteristic polynomial ? # 4 b) Compute the trace of B, the sum of diagonal terms of the matrix B in three ways. 1- directly, 2- via eigenvalues, 3-via characteristic polynomial # 4 c) Compute the determinant of B, ad — bc,...
2. 15 points Consider the following matrices 2 -2 2 3 -3 3 4 -2 2 4 6 2 12 2 2 36 1 1 18 3 3 53 -3 1 -22 2 21 33 -1 1 -1 1 ,ws 3 3 -2 3 -1 0 -20 0 -1 3 7 -3 2 Let V span^v1, v2, v3) and W-span{w1, w2, w3, wa,w5, ws). (a) By finding more suitable bases, give a simple description of the subspaces V and W....