Compute the following change of basis matrices
We will consider the vector space P2 with the bases B = ⟨x 2 + 2, 1 + x, 1 + 3x − 2x 2 ⟩ and D = ⟨1 + 2x + 2x 2 , 1 + x + x 2 , 1 + x − x 2 ⟩ As well as the vector space R 4 with the basis E = ⟨ 1 0 1 2 , 0 1 1 1 , 0 0 1 1 , 0 0 0 −1 ⟩ Exercise 1. Compute the follow change of basis matrices: 1. The change of basis matrix from B to D. 2. The change of basis matrix from D to B. 3. The change of basis matrix from E to the standard basis of R 4 . 4. The change of basis matrix from the standard basis of R 4 to E.
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(1 point) Consider the ordered bases B = (1 – X,4 – 3x) and C =...
(1 point) Consider the ordered bases B = (1 – X,4 – 3x) and C = (-(3 + 2x), 4x – 2) for the vector space P2[x]. a. Find the transition matrix from C to the standard ordered basis E = (1, x). -3 2 TE = -2 b. Find the transition matrix from B to E. 1 -1 T = 4 -3 c. Find the transition matrix from E to B. -3 1 T = 4/7 -1/7 d. Find...
(1 point) Consider the ordered bases B = {-(7 + 3x), –(2+ x)} and C =...
(1 point) Consider the ordered bases B = {-(7 + 3x), –(2+ x)} and C = {2,3 + x} for the vector space P2. a. Find the transition matrix from C to the standard ordered basis E = {1,x}. TE = b. Find the transition matrix from B to E. Te = c. Find the transition matrix from E to B. 100 TB = d. Find the transition matrix from C to B. TB = 11. !!! e. Find the...
1 (8 pts) Find the dimension and a basis for the following vector spaces. (a) (4...
1 (8 pts) Find the dimension and a basis for the following vector spaces. (a) (4 pts) The vector space of all symmetric 2 x 2 matrices (which is a subspace of M22). (b) (4 pts) All vectors of the form (a, b, 2a +36) (which is a subspace of Re"). 2. (12 pts) Given the matrix in a R R-E form: -21 1 [1 0 0 0 3 0 1 1 0 - 2 0 0 0 1 0...
1. Given matrices (A), [B][C], [D], and [E] compute the following. You must show all the...
1. Given matrices (A), [B][C], [D], and [E] compute the following. You must show all the steps in your solution to receive credit. (2 points each) -1 3 1] [-2 1 [-2 4 1 [A]*[B] [B]*[C] [D]*[A] [C*E] det[B] det[D] My of [D] M22 of [D] M21 of [D] Write the following system of equations in matrix form x-5y+32-6=0 3y +2x = -11 3x-7z+15= 0
Problem 4. Let GL2(R) be the vector space of 2 x 2 square matrices with usual matrix addition and...
Problem 4. Let GL2(R) be the vector space of 2 x 2 square matrices with usual matrix addition and scalar multiplication, and Wー State the incorrect statement from the following five 1. W is a subspace of GL2(R) with basis 2. W -Ker f, where GL2(R) R is the linear transformation defined by: 3. Given the basis B in option1. coordB( 23(1,2,2) 4. GC2(R)-W + V, where: 5. Given the basis B in option1. coordB( 2 3 (1,2,3) Problem 5....
no calculator please 1 (8 pts) Find the dimension and a basis for the following vector...
no calculator please
1 (8 pts) Find the dimension and a basis for the following vector spaces. (a) (4 pts) The vector space of all symmetric 2 x 2 matrices (which is a subspace of M22). (b) (4 pts) All vectors of the form (a, b, 2a + 3b) (which is a subspace of R®). 2. (12 pts) Given the matrix in a R R-E form: 1000 3 0110-2 00011 0 0 0 0 0 (a) (6 pts) Find rank(A)...
Problem 5 (25 points). Let Mat2x2(R) be the vector space of 2 x 2 matrices with...
Problem 5 (25 points). Let Mat2x2(R) be the vector space of 2 x 2 matrices with real entries. Recall that (1 0.0 1.000.00 "100'00' (1 001) is the standard basis of Mat2x2(R). Define a transformation T : Mat2x2(R) + R2 by the rule la-36 c+ 3d - (1) (5 points) Show that T is linear. (2) (5 points) Compute the matrix of T with respect to the standard basis in Mat2x2 (R) and R”. Show your work. An answer with...
(1 point) Find a basis for the column space of 0 A = -1 2 3...
(1 point) Find a basis for the column space of 0 A = -1 2 3 3 - 1 2 0 - 1 -4 0 2 Basis = (1 point) Find the dimensions of the following vector spaces. (a) The vector space RS 25x4 (b) The vector space R? (c) The vector space of 6 x 6 matrices with trace 0 (d) The vector space of all diagonal 6 x 6 matrices (e) The vector space P3[x] of polynomials with...
7. [2p] (a) In a two-dimensional linear space X vectors el, e2 formi a basis. In this basis a vector r E X has expansio...
7. [2p] (a) In a two-dimensional linear space X vectors el, e2 formi a basis. In this basis a vector r E X has expansion x = 2e1 + e2. Find expansion of the vector x in another basis 1 -2 er, e2, of X, if the change of basis matrix from the basis e to the basis e, s (b) In a two-dimensional linear space X vectors el, e2 forn a basis. In this basis a vector r E...
1. Let T: R2 – R? be the map "reflection in the line y = x"—you...
1. Let T: R2 – R? be the map "reflection in the line y = x"—you may assume this T is linear, let Eº be the standard basis of R2 and let B be the basis given by B = a) On the graph below, draw a line (colored if possible) joining each of the points each of the points (-). (). (1) and () woits image to its image under the map T. y = x b) Find the...
(1 point) Consider the ordered bases B = (1 – X,4 – 3x) and C = (-(3 + 2x), 4x – 2) for the vector space P2[x]. a. Find the transition matrix from C to the standard ordered basis E = (1, x). -3 2 TE = -2 b. Find the transition matrix from B to E. 1 -1 T = 4 -3 c. Find the transition matrix from E to B. -3 1 T = 4/7 -1/7 d. Find...
(1 point) Consider the ordered bases B = {-(7 + 3x), –(2+ x)} and C = {2,3 + x} for the vector space P2. a. Find the transition matrix from C to the standard ordered basis E = {1,x}. TE = b. Find the transition matrix from B to E. Te = c. Find the transition matrix from E to B. 100 TB = d. Find the transition matrix from C to B. TB = 11. !!! e. Find the...
1 (8 pts) Find the dimension and a basis for the following vector spaces. (a) (4 pts) The vector space of all symmetric 2 x 2 matrices (which is a subspace of M22). (b) (4 pts) All vectors of the form (a, b, 2a +36) (which is a subspace of Re"). 2. (12 pts) Given the matrix in a R R-E form: -21 1 [1 0 0 0 3 0 1 1 0 - 2 0 0 0 1 0...
1. Given matrices (A), [B][C], [D], and [E] compute the following. You must show all the steps in your solution to receive credit. (2 points each) -1 3 1] [-2 1 [-2 4 1 [A]*[B] [B]*[C] [D]*[A] [C*E] det[B] det[D] My of [D] M22 of [D] M21 of [D] Write the following system of equations in matrix form x-5y+32-6=0 3y +2x = -11 3x-7z+15= 0
Problem 4. Let GL2(R) be the vector space of 2 x 2 square matrices with usual matrix addition and scalar multiplication, and Wー State the incorrect statement from the following five 1. W is a subspace of GL2(R) with basis 2. W -Ker f, where GL2(R) R is the linear transformation defined by: 3. Given the basis B in option1. coordB( 23(1,2,2) 4. GC2(R)-W + V, where: 5. Given the basis B in option1. coordB( 2 3 (1,2,3) Problem 5....
no calculator please
1 (8 pts) Find the dimension and a basis for the following vector spaces. (a) (4 pts) The vector space of all symmetric 2 x 2 matrices (which is a subspace of M22). (b) (4 pts) All vectors of the form (a, b, 2a + 3b) (which is a subspace of R®). 2. (12 pts) Given the matrix in a R R-E form: 1000 3 0110-2 00011 0 0 0 0 0 (a) (6 pts) Find rank(A)...
Problem 5 (25 points). Let Mat2x2(R) be the vector space of 2 x 2 matrices with real entries. Recall that (1 0.0 1.000.00 "100'00' (1 001) is the standard basis of Mat2x2(R). Define a transformation T : Mat2x2(R) + R2 by the rule la-36 c+ 3d - (1) (5 points) Show that T is linear. (2) (5 points) Compute the matrix of T with respect to the standard basis in Mat2x2 (R) and R”. Show your work. An answer with...
(1 point) Find a basis for the column space of 0 A = -1 2 3 3 - 1 2 0 - 1 -4 0 2 Basis = (1 point) Find the dimensions of the following vector spaces. (a) The vector space RS 25x4 (b) The vector space R? (c) The vector space of 6 x 6 matrices with trace 0 (d) The vector space of all diagonal 6 x 6 matrices (e) The vector space P3[x] of polynomials with...
7. [2p] (a) In a two-dimensional linear space X vectors el, e2 formi a basis. In this basis a vector r E X has expansion x = 2e1 + e2. Find expansion of the vector x in another basis 1 -2 er, e2, of X, if the change of basis matrix from the basis e to the basis e, s (b) In a two-dimensional linear space X vectors el, e2 forn a basis. In this basis a vector r E...
1. Let T: R2 – R? be the map "reflection in the line y = x"—you may assume this T is linear, let Eº be the standard basis of R2 and let B be the basis given by B = a) On the graph below, draw a line (colored if possible) joining each of the points each of the points (-). (). (1) and () woits image to its image under the map T. y = x b) Find the...
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