)5. The operator T on a vector space V over Q has as its sequence of invariant factors the sequence (x-1)3(x-2)5(x-3)2, (x-1)(x-2)(x-3)2, (x-3). Find the Jordan normal form of T.

Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors {vu,wv,uw} is linearly independent or linearly dependent.

3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.

Let P3 be the vector space of all polynomials of degree 3 or less in the variable z. Let = 2+x+x², 2+x+x², 2+x², = 11 + 3x + 6x² choose PI(T) P2(x) P3(x) = P4(x) and let C = {p1(x), p2(x), P3(x), P4(x)}. a. Use coordinate representations with respect to the basis B = {1, 2, ², ³} to determine whether the set C forms a basis for P.. = c. The dimension of span(C) is b. Find a basis for span(C). Enter a polynomial or a comma separated list of polynomials. {}

3. [6 points] Consider the vector space V = P2 over the field of real numbers F = R. Is the vector 4 – x – x² in the span of the vectors 1+ 2x, x – x², -1+x+x² Explain why or why not. Show your calculations.

(3) For each of the following sets, determine whether it is linearly independent or dependent. (0) € 9000 3 (c) {x³x, 2x² + 4,-2x³ +3x²+2x+6} in P₂ (R), where P3 (R) denotes the vector space of polynomials over the reals of degree at most 3. -1 2 (d) {(-12 1) (1 7¹). (1¹3) (3¹2)} in M2x2(R), where M2x2(R) 0 denotes the vector space of 2 × matrices over the reals. {0·0)} span(S), i.e., orthogonal to every vector in span(S). (b) (4) Let S in R4. 7 in R4. CR4. Find all vectors in R4 that are orthogonal to (5) Let A and B be matrices over the reals. For each of the following statements, determine whether it is true or false. If it is true, prove it. If it is false, give a counter example to disprove it. (a) If A + B is defined, then rank(A + B) = rank(A) + rank(B). (b) If AB is defined, then rank(AB) = rank(A) rank(B). (c) If A has size m × n, then rank(A) ≤ min{m, n}. (d) rank(A) = rank(At) (e) nullity(A) = nullity(A¹)