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APPM 3310: Matrix Methods — Exam #1 — June 16, 2009 On the front of your bluebook print (1) your name, (2) your student ID number, and (3) a grading table. Explain all of your answers. A correct answer with no supporting work may receive no credit while an incorrect answer with some correct work may receive partial credit. No electronic devices of any kind (e.g. cell phones, calculators, etc.) are permitted. The test is worth 100 points, however there are 110 possible points available, allowing for up to 10 extra credit points. 1, (10 points) State the Fundamental Theorem of Linear Algebra, part 1. Include a careful definition of each of the four fundamental subspaces associated with an m x n matrix A 2. (30 points) For this problem, let 2 1 0 5 A= 4 2-1 8 ~-2 -1 8 -4 yz. (a) Find the LU decomposition of A where U is in row echelon form. & (b) Determine the rank of A and the dimensions of the four fundamental subspaces associated with A. (@ (c} Find a basis for ker(A) and mmg(A). fo (d) Write down the general form for the space of solutions to the homogeneous system Ax = 0. 3. (30 points) For each of the following, if you are asked to prove something, simply provide a proof, For True/False questions, remember that True answers require a complete proof for the general case and False answers require a counterexample. Ayn | lo (a) Prove that if A and B are n x n matrices then trAB = trBA. Anu Awe v e (b) True or False Every square matrix A commutes with its transpose. Alncn. ® (c) True or False A set of vectors is linearly dependent if the zero vector belongs to their span. & {d) True or False Given two n x n matrices A and B if ker(A) = ker(B) then rank(A) = rank(B). (g (©) Prove that the set of {A € MaxnltrA = 0} is a subspace of Maxn 4. (20 points) Let P“) denote the vector space of all polynomials p(x) of degree less than or equal to 4. VS (a) Are pi(x) = 2? ~ 82 +1, pola) x* — 6x +3, p3(2) = «4 — 2x + 1, linearly independent elements of P(4)? § (b) What is the dimension of the subspace of P spanned by pi, po. ps? 5. (20 points) Let v1, vo, ..., Ve € R” and let A = (vq vy) be the corresponding matrix whose columns are the v;. Prove that the columns of A are linearly dependent if and only if there is a : non-zero solutions ¢ # 0 to the homogeneous lincar system Ac = 0. 
