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M346 First Midterm Exam, September 21, 2004 1. Let V be the subspace of R4 defined by the equation x1 + x2 + x3 + x4 = 0. a) Find the dimension of V . b) Find a basis for V . [Any basis will do, but the simpler your answer, the easier part (c) will be. Be sure that each of your vectors really is in V , and that they are linearly independent] x2 x 3 c) Let L(x) = . Note that L takes V to V , and can be viewed as an x4 x1 operator on V . Find the matrix [L]B , where B is the basis you found in part (b). µ ¶ µ ¶ 5 3 2 2. In R , consider the basis b1 = , b2 = . 3 2 a) Find the change-of-basis matrices PEB and PBE , where E is the standard basis. µ ¶ 13 b) If v = , find [v]B . −2 ¶ ¶ µ µ 2x2 x1 . Find [L]E and [L]B . = c) Let L x1 + x 2 x2 3. Consider the coupled first-order differential equations dx1 = x1 +2x2 dt dx2 = 2x1 + x2 dt Define the new variables y1 (t) = x1 (t) + x2 (t), y2 (t) = x1 (t) − x2 (t). a) Rewrite the system of equations completely in terms of y1 and y2 . (That is, express dy1 /dt and dy2 /dt as functions of y1 and y2 .) b) Given the initial conditions x1 (0) = 1, x2 (0) = 0, find x1 (t) and x2 (t). 4. Let V =R3 [t], and let L : V → V be defined by L(p)(t) = p0 (t) + 2p00 (t). a) Find [L]E , where E = {1, t, t2 , t3 } is the standard basis. b) What is the dimension of the kernel of L? What is the dimension of the range of L? c) Find a basis for the kernel of L. d) Find a basis for the range of L. 1 5. True of False? Each question is worth 4 points. You do NOT need to justify your answers, and partial credit will NOT be given. 1 0 2 0 0 1 1 0 . For (a) and (b), suppose that a 4×4 matrix A row-reduces to 0 0 0 1 0 0 0 0 T a) The null space of A is the span of (−2, −1, 1, 0) . 1 0 0 0 1 0 b) The column space of A is the span of , , and . 0 0 1 0 0 0 For (c) and (d), suppose that L :R2 [t] → M2,2 is a linear transformation, and that B = [L]EE is the matrix of L relative to the standard bases for R2 [t] and M2,2 . c) If B row-reduces to something with 3 pivots, then L is 1–1. 1 µ ¶ 3 1 3 is in the column space of B. d) If is in the range of L, then 4 4 7 7 3 e) R is the internal direct sum of the x1 -x2 and x1 -x3 planes. 2