Problems: 1. (6 points) Consider the following list of elements in P3(F): 2241, 22z 242 242z2+1 (a) Prove that these elements are linearly independent. (b) Based on part (a) and the dimension of P3(F), do these elements span? Please explain. 2. (8 points) In each case below, find a basis for the given vector space V over F. Please prove that your answer is a basis. (a) F:CandV:{QECE' : Ag:g}, where 1 2 0 0 = A=12 25 2 4 2 3 3 1 2 43 Hint: Start by solving this system of equations in vector form. (b) F=Rand V ={A My(C) : AB = BA}, where 0 1 o=l b : : : 4l and then see what constraints the equation AB = BA imposes on the entries a,b,c,d C. Hint: Write A = 'C' Problems: 1. (3 points) Define T : F3 -> F2 by a - 4c 26 + 3c for any E F3 Prove that T is a linear transformation. 2. (3 points) Define T : P(F) - P(F) by the rule T(p(z)) = zp(z), for any p(z) EP(F). Prove that T is a linear transformation. 3. (5 points) Consider the vector space P2(F), with its elements fi(z) = 22 +1, f2 ( z) = 22 + z -1, f3(2) = -2z+2 Since these form a basis for P2(F), there is a linear transformation T : P2(F) + M2(F) completely determined by the requirement that T(f1 (2)) = 10 8, I(f2 ( 2) = 18 0, I( f3 ( 2 ) - 1 1 Determine the values T(1), T(z) and T(z2)