Math 203 Practice Exam 2 Please work out each of the given problems.  Credit will be based on the steps towards the final answer.  Show your work.  Do your work on your own paper. Problem 1 Let  S  =  {t2, t2 + 2t, t2 + 3}  and T  =  {2t - 1, 5t - 3, t2} be subsets of P2  A.     Prove that S is a basis for P2.   B.     Find the change of basis matrix PS<--T.     Problem 2 Of the following two subsets of the vector space of differentiable functions, determine which is a subspace.  For the one that is not a subspace, demonstrate why it is not.  For the one that is a subspace, prove that it is a subspace. A.  S  =  {f | f(3)  =  f '(3)}  B.  T  =  {f | f(0)f '(0)  =  0}    Problem 3 Let V be the subspace of differentiable functions spanned by {ex, e2x, e3x} and let         L:  V --->  V be the linear transformation with          L(f(x))  =  f ''(x) - 3f '(x) + 2f(x)   A.     Write down the matrix AL with respect to the given basis. B.     Find the a basis for the kernel and range of L.   Problem 4  Let W = Span{(1,1,0,1), (0,1,2,3)}.  Find a basis for the orthogonal complement of W. Problem 5 Let Find the following: A basis for the column  space of A. A basis for the row space of A. A basis for the null space of A. The nullity of A. The rank of A.    Problem 6 Let  L:  V ---> V  be a linear transformation.  Use the fact that          dim(Ker L) + dim(Range L)  =  dim(V) to show that if L is one to one then L is onto.  Problem 7  Let A and B be matrices and let v be an eigenvector of both A and B.  Prove that v is an eigenvector of the product AB. Problem 8 Without the use of a calculator, diagonalize the matrix   Problem 9 Answer True of False and explain your reasoning.   A.    Let A be a 3x3 matrix such that the columns of A form an orthonormal set of vectors.  Then           B.  If A is an mxn matrix such that the rows of A and the columns of A are both linear independent sets, then n = m.C.  If u and v are eigenvectors for a matrix A, then u + v is also an eigenvector for A. D.  If A is similar to B and B is similar to C, then A is similar to C. E.  If l is an eigenvalue for A, then l2 is an eigenvalue for A2. F.  The vectors (1,1,0), (1,0,1), and (0,0,1) are a basis for P2.