Math 203 Practice Final Exam Please work out all of the following problems.  Credit will be given based on the progress that you make towards the final solution.  Show your work.  No calculators allowed for this page. Printable Key Problem 1  Let          Find an orthogonal matrix P and a diagonal matrix D with A  =  PDP-1. Problem 2 Use the permutation definition of the determinant to find the determinant of           Problem 3 Find the inverse of A if           Calculators are permitted on this part   Problem 4 Consider the matrix              A.     Determine the rank of A. Solution B.   Find a basis for the null space of A. Solution C.     Find a basis for the column space of A using columns of A. Solution Problem 5       Let    be defined by         A.    Prove that L is a linear transformation. B.   Let S = {x2 + x, x2 + 1, x} and T = {(1,1), (1,2)} be bases for P2 and R2.  Find the matrix for L with the bases S and T.    Problem 6  Show that the set         is a basis for M2x 2.   Problem 7 Use matrices to find the unknown currents in the given circuit.           Problem 8 Graph the equation and write the equation in standard form.         4x2 + 2xy + 4y2  =  15 Problem 9 One of the following is a subspace of the space of differentiable functions.              I.   {f | f(0) – f ‘(0)  =  1}          II. {f | f(1)  =  f ‘(1)}    A.     Determine which is not a subspace and explain why. B.     Prove that the other one is a subspace.    Problem 10 Prove that if A is a matrix such that A2 = 0 then 0 is an eigenvalue for A. Problem 11 Answer the following true or false. If it is true, explain why.  If it is false explain why or provide a counter example. A.    If S  =  {v1, v2} is a linearly independent set of vectors in R3 and v3 is not in the span of S, then  {v1, v2, v3} is a basis for R3. B.     Every orthonormal set of five vectors in R5 is a basis for R5. Solution C.     Let A and B be matrices such that A2v = a, B2v = b, and ABv = c.  Then (A + B)2 v  =  a + b + 2c      Solution