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U 2 A) 4 d vector space V such that -001 - 402. Find the change-of-coordinates...
(6) In each case V is a vector space, T: V- V is a linear transformation,...
(6) In each case V is a vector space, T: V- V is a linear transformation, and v is a vector in V. Determine whether the vector v is an eigenvector of T If so, give the associated eigenvalue Is v an eigenvector? If so, what is the eigenvalue? (b) T : M2(R) → M2(R) is given by [a+2b 2a +b c+d2d and V= Is v an eigenvector? If so, what is the eigenvalue? (c) T : R2 → R2,...
7.3 (Eigenvalues II) Let V be a vector space over K and let f,g E End(V)....
7.3 (Eigenvalues II) Let V be a vector space over K and let f,g E End(V). Show that: a) If-1 is an eigenvalue of ff, then 1 is an eigenvalue of f3. b) If u is an eigenvector off o g to the eigenvalue λ such that g(v) 0, then g(v) is an eigenvector of g o f. If, in addition, dim V < oo,then f o g and go f have the same eigenvalues c) If {ul, unt is...
Let v 2 Rn be a unit vector. Define G = I ? vvT . (a)...
Let v 2 Rn be a unit vector. Define G = I ? vvT .
(a) Show G is symmetric and G2 = G.
(b) Prove v is an eigenvector, find the associated
eigenvalue.
(c) Prove that if < u; v >= 0 then u is also an eigenvector
of G.
(d) Prove that G is diagonalizable.
Let v ER" be a unit vector. Define G=I - vt. (a) Show G is symmetric and G =G. (b) Prove v is...
3. Given that 8 - ...) is a basis for a vector space V. Determine if...
3. Given that 8 - ...) is a basis for a vector space V. Determine if 3 - + - +213 + 3) is also a for V 9. Find the change of coordinates matrix P from the basis B = {1 + 21,2 + 3t) to the basis C = {1,1+56) of P,
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue...
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
Given the matrix A= 76 -2 -4 -4 8 8 1 4 -4 -4 X =...
Given the matrix A= 76 -2 -4 -4 8 8 1 4 -4 -4 X = 2 is an eigenvalue of A and 12 = 4 is an eigenvalue of A of multiplicity 2. (a) Find the eigenvector(s) corresponding to l1 = 2. (b) Find the eigenvector(s) corresponding to 12 = 4. (C) Find the general solution of x' = Ax.
(1 point) Find the characteristic polynomial of the matrix 5 -5 A = 0 [ 5...
(1 point) Find the characteristic polynomial of the matrix 5 -5 A = 0 [ 5 -5 -2 5 0] 4. 0] p(x) = (1 point) Find the eigenvalues of the matrix [ 23 C = -9 1-9 -18 14 9 72 7 -36 : -31] The eigenvalues are (Enter your answers as a comma separated list. The list you enter should have repeated items if there are eigenvalues with multiplicity greater than one.) (1 point) Given that vi =...
2 (5 points) Recalled that null space of a matrix A € Mnxn is defined as...
2 (5 points) Recalled that null space of a matrix A € Mnxn is defined as N(A) = {r € R” : Ar =0}. Now, the eigenspace of A corresponding to the eigenvalue 1 (denoted by Ex(A)) is defined as the nullspace of A-XI, that is, EX(A) = N(A – XI) = {v ER”: (A – XI)v = 0}. You should have three distinct eigenvalues in Problem 1 above. Let say there are li, 12, and 13. (i) Find the...
Consider the differential equation for the vector-valued function x, x = x, A- Find the eigenvalues...
Consider the differential equation for the vector-valued function x, x = x, A- Find the eigenvalues A, , and their corresponding eigenvectors V, V, of the coefficient matrix A (a) Eigenvalues Au, dy (b) Eigenvector for A, you entered above (c) Eigenvector for A, you entered above: V2 = (d) Use the eigenpairs you found in parts (a)-(C) to find real-valued fundamental solutions to the differential equation above X = X Note: To enter the vector (u, v) type <u,v>...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ]...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
(6) In each case V is a vector space, T: V- V is a linear transformation, and v is a vector in V. Determine whether the vector v is an eigenvector of T If so, give the associated eigenvalue Is v an eigenvector? If so, what is the eigenvalue? (b) T : M2(R) → M2(R) is given by [a+2b 2a +b c+d2d and V= Is v an eigenvector? If so, what is the eigenvalue? (c) T : R2 → R2,...
7.3 (Eigenvalues II) Let V be a vector space over K and let f,g E End(V). Show that: a) If-1 is an eigenvalue of ff, then 1 is an eigenvalue of f3. b) If u is an eigenvector off o g to the eigenvalue λ such that g(v) 0, then g(v) is an eigenvector of g o f. If, in addition, dim V < oo,then f o g and go f have the same eigenvalues c) If {ul, unt is...
Let v 2 Rn be a unit vector. Define G = I ? vvT .
(a) Show G is symmetric and G2 = G.
(b) Prove v is an eigenvector, find the associated
eigenvalue.
(c) Prove that if < u; v >= 0 then u is also an eigenvector
of G.
(d) Prove that G is diagonalizable.
Let v ER" be a unit vector. Define G=I - vt. (a) Show G is symmetric and G =G. (b) Prove v is...
3. Given that 8 - ...) is a basis for a vector space V. Determine if 3 - + - +213 + 3) is also a for V 9. Find the change of coordinates matrix P from the basis B = {1 + 21,2 + 3t) to the basis C = {1,1+56) of P,
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
Given the matrix A= 76 -2 -4 -4 8 8 1 4 -4 -4 X = 2 is an eigenvalue of A and 12 = 4 is an eigenvalue of A of multiplicity 2. (a) Find the eigenvector(s) corresponding to l1 = 2. (b) Find the eigenvector(s) corresponding to 12 = 4. (C) Find the general solution of x' = Ax.
(1 point) Find the characteristic polynomial of the matrix 5 -5 A = 0 [ 5 -5 -2 5 0] 4. 0] p(x) = (1 point) Find the eigenvalues of the matrix [ 23 C = -9 1-9 -18 14 9 72 7 -36 : -31] The eigenvalues are (Enter your answers as a comma separated list. The list you enter should have repeated items if there are eigenvalues with multiplicity greater than one.) (1 point) Given that vi =...
2 (5 points) Recalled that null space of a matrix A € Mnxn is defined as N(A) = {r € R” : Ar =0}. Now, the eigenspace of A corresponding to the eigenvalue 1 (denoted by Ex(A)) is defined as the nullspace of A-XI, that is, EX(A) = N(A – XI) = {v ER”: (A – XI)v = 0}. You should have three distinct eigenvalues in Problem 1 above. Let say there are li, 12, and 13. (i) Find the...
Consider the differential equation for the vector-valued function x, x = x, A- Find the eigenvalues A, , and their corresponding eigenvectors V, V, of the coefficient matrix A (a) Eigenvalues Au, dy (b) Eigenvector for A, you entered above (c) Eigenvector for A, you entered above: V2 = (d) Use the eigenpairs you found in parts (a)-(C) to find real-valued fundamental solutions to the differential equation above X = X Note: To enter the vector (u, v) type <u,v>...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
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