Let F be a field, and let A be the following 4 × 4 matrix over F: 0 0 -1 0 0 A = -1 1 0 -1 0 0 1 -1 (i) Find the characteristic polvnomial of A. (ii) Find the minimal polynomial of A. For which of the following fields F is A diagonalisable over F: (iii) (a) F = C? (B) F = R? (7) F = F3, the field of 3 elements? %3| %3D

T: V ---> V is a linear operator with a B and B' ordered bases for V a. Find the matrix representation for T relative to the ordered bases B and B' b. Find T(v), using a direct computation and using the matrix representation

2. Prove Cayley-Hamilton Theorem for a diagonalizable matrix, A: Let A be diagonalizable and p(x) be its characteristic polynomial; i.e. value of A. p(A) : = 0 for A, eigen- Show that p(A) = 0.

Suppose S: R² → R² and T: R² → R² are linear transformations given by 3x y (0)-[-1-(C)-[22] T -x-3y] (a) Find the standard matrix for S (assuming the stand basis for R³ and for R²). (b) Find the standard matrix for T (assuming the standard basis for R² and for R³). (c) Show that S and T are invertible. (d) Show that T is the inverse of S. (e) What is the standard matrix for T(S(z)) = To S(1), where I = R²?

4. Let E: P3 R2 be given by E (p(x)) = [P(2)] (a) Find a matrix that induces the transformation E. TH (b) Find a polynomial p(z) where E(p(x)) = [], or explain why this isn't possible. (c) Find a polynomial q(2) where E(q(z)) = [], or explain why this isn't possible. 8P (d) Find a nonzero polynomial r(z) where E(r(2)) = [] or explain why this isn't possible. (e) Describe the kernel of E. What is the dimension of the kernel? Find a basis for the kernel. (f) Describe the image of E. What is the dimension of the image? Find a basis for the image. (g) Is the transformation E one to one? Is it onto? Is it an isomorphism? Explain.

IV. Let L be the operator given by L[y] = y"" – 2y" + 2u" i. Consider L as a linear operator T : P3 → P3. Find [T], where a is the standard basis of P3 ii. Determine the kernel of L from part i. iii. Now, consider L as a transformation L: Co (R) → C (R). Find the kernel of L.

A Let T be the linear transformation of P2 (F) defined by the formula T(P(x)) = (x + 2)P'(x) − P(x) a) Find the matrix of T in the standard basis (1, x, x²). :

Let A be the matrix defined by [4 1 A = 0 -3 Let T : R? → R² be the map defined by T(x) = Ax. (2, –3). Compute A10(x) by using the matrix (b) M(T) above. Don't forget to convert x to the basis B, and then Let x = convert back.