Mathematics 54W Professor A. OgusFall, 2007Sample Midterm 2Work each problem on a separate sheet of paper. Be sure to put your name,your section number, and your GSI’s name on each sheet of paper. Also,at the top of the page, in the center, write the problem number, and besure to put the pages in order. Write clearly: explanations (with completesentences when appropriate) will help us understand what you are doing.Note that there are problems on the back of this sheet, for a total of fiveproblems.1. Let A :=1 10 01 0, let W be the column space of A, and letY :=113.(a) Use the Gram-Schmidt process to find an orthogonal basis for W .(b) Find the orthogonal projection Y0of Y on W .(c) Find the distance from Y to W .(d) Find X such that ATAX = ATY . (Hint: use part (a) to savesome work.)2. Let A :=54 81−9 0.(a) Find the characteristic polynomial of A.(b) Find the eigenvalues of A.(c) Find a diagonal matrix D and a nilpotent matrix N such thatA = D + N.(d) Use part (a) to find a matrix B such that that B3= A.3. Let A :=5 12−2 −5.(a) Find the characteristic polynomial of A.(b) Find the eigenvalues of A.(c) Find an invertible matrix S and a diagonal matrix D such thatA = SDS−1.(d) List all possibilities for D. Explain.(e) Find a matrix B such that B3= A.4. Let A by an n × n matrix whose only eigenvalue is 0.(a) Is A necessarily 0? Give a proof (explanation) or counterexample.You may use a theorem proved in class.(b) Answer the same question, assuming now that A is symmetric.5. Let V be the subspace of the space of functions R → R spanned byB := (x sin x, x cos x, sin x, cos x).(a) Show that the the differentiation operator D sends V to itself.(b) Show that the sequence B is linearly independent.(c) Find the matrix for D with respect to the basis B.(d) Find the eigenvectors and eigenvalues of D.(e) Is D diagonalizable?
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