Let S (2,0, 1), 2- (1,2,0),s (1, 1, 1)) and J- (w (6,3,3), w (4,-1,3),u3 (5,5,...
Tk 1 21 5 -5 k (a) Find the determinant of A in terms of k (b) For which value(s) of k is the matrix A invertible? (c) Let B-(k,1,2,0), (0, k, 2,0),(5,-5, k,0)) be a set of vectors in R4, and let k equal some answer you gave for part (b) of this question. Add an appropriate number of vectors to B so that the resulting set is a basis for R'
Tk 1 21 5 -5 k (a)...
6 The use of cell phones, laptogs, and Poas s prohibited during the examination EXERCISE 1 [2.5/10] a) [1/10] Let &-((01,-1).(.1,). (1,0,1)) be a basis of R. Calculate the coordinates Of the vector 굿= +ē2 with respect to the basis B. (民=(6.G.G} is the canonical basis) [1,5/10] LetBe闻珂司andB"=(utuatust) be two bases of R3, where: b) Calculate the change of basis matrix from & to &. EXERCISE 2 (2.5/10] Given the following vector subspaces:
6 The use of cell phones, laptogs,...
In the vector space R, let 8 {(1,3,0), (1, -3, 0), (0, 2, 2)}. (a) (6 points) Show that y is a basis of R3. (b) (7 points) Find the matrix [I,where I is the identity transform R3 R3 (c) (7 points) Using the matrix [I, convert the vector (r, y, z) into coordinates with respect to y instead of B. In other words, find ((x, y, z)] {(1,0,0), (0, 1,0), (0,0, 1)} be the standard basis, and let
linear algebra
Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and (a) Prove: V is a basis. (b) Find the coordinates of (b, b2, bs) with respect to V = {71, U2, 3,}. (c) Suppose M and M' are matrices whose columns span the same vector space V. Let b be the coordinates of relative to M. Write a matrix equation that gives b', the coordinates of relative to M'. (Your answer should be a...
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could you post solutions to the following questions. Thanks.
2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 state whether they are subepaces of R3 or not by clearly explaining your answer. 2% 2% (c) Consider the map F : R2 → R3 defined by for any z = (zi,Z2) E R2. 3% 3% 3% 3% i. Show that...
With explanation!
3. Let B2 be the linear operator B2f (x):- f(0)2 2 (1f (1)2, which maps functions f defined at 0, 1 to the quadratic polynomials Pa. This is the Bernstein operator of degree 2, Let T = B21Py be the restriction of B2 to the quadratics. (a) Find the matrix representation of T with respect to the basis B = [1,2,2 (b) Find the matrix representation of T with respect to the basis C = (1-x)2, 22(1-2),X2]. (c)...
Prove the following: (a) Let V be a vector space of dimension 3 and let {v,U2,U3} be a basis for V. Show that u2, u2 -2+s and uvi also form a basis for V (b) Show that1-,1-2,1-- 2 is a basis for P2[r], the set of all degree 2 or less polynomial functions. (c) Show that if A is invertible, then det A (Note: Show it for any det A-1 square matrix, showing it for a 2 x 2 matrix...
71121 4 1 15 Let S = -21,-1, 3 and C = | 1,2], o be bases for R'. Find the change-of- L5 JL-1] [4] IL-5] [ 8 7 ] coordinates matrix from B to C and the change-of-coordinates matrix from C to B. Find [x]. for x, = 4b, - 2b, +3b,. Find [x], for x, = 4c, - 2c, + 3e, Show these work by finding the coordinates of each vector in the standard basis using Band C
(1 point) Let S = {1, 2, 3} and T : Fun(S) + Rº be the transformation T(f) = (f(2) – 2 f(1), f(2) + f(3), f(1)) and consider the ordered bases E = {x1 X1, X2, X3 > the standard basis of Fun(S) F = {xı – X3, 2X1 + X2, X3 – x2} a basis of source Fun(S) E' = {(1,0,0), (0,1,0), (0,0,1)}the standard basis of R3 G = {(-2, –1,1), (1,-1,0), (0,1,0)} a basis of target R3...
2. Let e1(x) = 1, ez(x) = x, p1(x) = 1 – x and p2(x) = –2 + x. Let E = (e1,e2) and B = (P1, P2). 2 a) Show that B is a basis for P1(R). 4 b) Let ce : R → R3 be the change of coordinates from E to ß. Find the matrix representation of C. Leave your answer as a single simplified matrix. 6 c) Let (:,:) be an inner product on P1(R). Suppose...