For each of the following operators T on a vector space V, find an ordered basis...
1 point) Read 'Diagonalization Changing to a Basis of Eigenvectors' before attempting this problem. Suppose that V is a 5-dimensional vector space. Let S -(vi,... , vs) be some ordered basis of V, and let T-(wi.... . ws) be some other ordered basis of V. Let L: V → V be a linear transformation. Let M be the matrix of L in the basis Sand et N be the matrix of L in the basis T. Decide whether each of...
2. Consider the inner product space V = P2(R) with (5,9) = . - f(t)g(t) dt, and let T:V + V be the linear operator defined by T(F) = xf'(x) + 2f (x). (i) Compute T*(1 + x + x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors ß for which [T]2 is diagonal. If such a basis exists, find one.
3. Consider the vector space V = R2[x] with its standard ordered basisE = 1,x,x2 and the linear map T :R2[x]−→R2[x], T(p)=p(x−1)−p(0)x2 (a) (1 point) What is [T]E? (b) (1 point) Is T invertible? (c) (6 points) Compute the eigenvalues of T and their algebraic multiplicity. (d) (2 points) Is T diagonalisable? If so, find a matrix Q such that Q−1[T]EQ is diagonal. If not, findQ, so that the above matrix is upper triangular.
Let S = {t2.t-1,1} be an ordered basis for P2(t). If the vector v in P2(6) has the coordinate vector 2 3 with respect to S, then what is the vector v? Select one: O at2 + 2t +1 O b. +2 +1+1 O c. 12 + 2t - 1 O d. t2 + 2t
2. Consider the inner product space V = P2(R) with (5.9) = £ 5(0)9(e) dt, and let T:V V be the linear operator defined by T(f) = xf'(2) +2f(x). (i) Compute T*(1+2+x²). (ii) Determine whether or not there is an orthonormal basis of eigenvectors 8 for which [T], is diagonal. If such a basis exists, find one.
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...
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> V be a linear transformation (a) Prove that for every pair of ordered bases B = (Ti,...,T,) of V and C = (Wi, ..., Wm) of W, then exists a unique (B, C)-matrix of T, written A = c[T]g. (b) For each n e N, let Pn be the vector space of polynomials of degree at mostn in the...
Determine whether or not the following transformation T :V + W is a linear transformation. If T is not a linear transformation, provide a counter example. If it is, then: (i) find the nullspace N(T) and nullity of T, (ii) find the range R(T) and rank of T, (iii) determine if T is one-to-one, (iv) determine if T is onto. : (a) T: R3 + R2 defined by T(x, y, z) = (2x, y, z) (b) T: R2 + R2...
(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...
7. Let V be the space generated by the basis B = {sin(t), cos(t), et}. i.e. V = span(B). Consider the linear transformation T:V + V defined by T(f(t)) = f"(t) – 2f'(t) – f(t). Find the standard matrix of the transformation. (Hint: Associate sin(t) with the vector (0), and so forth.) 8. Show that B = {t2 – 2, 3t2 +t, t+t+8} is a basis for P2, and find the change of coordinates matrix P which goes from B...