1 Linear Algebra Professor M. Zuker Mid-term Test; October 23, 2013. You may use the text book, posted lectures, the course web site, problem solutions and your notes. You may not work with others. Please write to me or the TA if a question is not clear or if you suspect an error. 1. In P(F), let p0(x) = 1, p1(x) = (x + 1), p2(x) = (x+1)(x+2) 2! and so on. In general, pi(x) = (x+1)(x+2):::(x+i) i! Let Wn = span(p0(x); p1(x); : : : ; pn(x)). What is dimWn? 2. Let T 2 L(R4) be dened by T(w; x; y; z) = (w + 2x + 3y + 4z; 5w + 6x + 7y + 8z; 9w + 10x + 11y + 12z; 13w + 14x + 15y + 16z). Compute A = Mat(T; (e1; e2; e3; e4)). Compute dim null T and a basis for null T. Then compute dim range T and a basis for range T. Hint: You should be able to compute a basis for range T just by glancing at the matrix A. 3. In this group of problems, you are asked to dene subspaces or linear maps that have certain properties. 1. Dene three subspaces U1, U2 and U3 of F10 such that dim U1 U2 U3 = 1, dim Ui Uj = 2 for all pairs of subspaces (for all i 6= j), dim U1 = dim U2 = dim U3 and F10 = U1 + U2 + U3. 2. Dene T 2 L(F8) such that null T = range T. Can such an operator exist in L(F7)? 3. Suppose that V = U1 + U2, where dim V = 15, dim U1 = dim U2 and dim U1 U2 = 3. Compute dim U1. 4. Let T 2 L(F6) and suppose that dim range T = 3 and that T has precisely 4 distinct eigenvalues, f1; 2; 3; 4g. Is it true that F6 =M4 i=1 null(T ?? iI)? 5. Suppose that C = BA, where A is a 2 3 matrix and B is a 3 2 matrix. Then C is a 3 3 matrix. Can C be invertible? If yes, give an example. If not, give a proof. 6. In Rn, let V = f(x1; x2; : : : ; xn) j xi + xi+1 = 0 for 1 i