Problems: 1. Consider the vector spaces V = P5(F) and W = M2,3 (F) over F. (a) (2 points) By considering only their dimensions, explain why V and W are isomorphic. (b) (3 points) Find an isomorphism T : V - W. Please justify your answer 2. Consider the plane V in R3 defined by the equation x + 2y + 3z = 0 (a) (1 point) Why is V a subspace of IR3? (b) (3 points) Find an isomorphism between R2 and V. 3. Consider the linear transformation T : P2(F) - P2(F) defined by d T(P(z) ) = p(z)+ 12 P ( z ) , where as per usual P2 (F) = {az2 + bz + c : a, b, cc F). (a) (3 points) Find a basis for Im(T). (b) (1 point) What is dimp Ker(T)? Hint: Use the Rank-Nullity Theorem. (c) (1 point) Is T an isomorphism? Please explain