Download MATH 51 Midterm I - October 19, 2006 and more Exams Calculus in PDF only on Docsity! MATH 51 MIDTERM I October 19, 2006 Instructions: • No calculators, books, notes, or electronic devices may be used during the exam. • You have 90 minutes. • There are 7 problems, each with multiple parts. You should work quickly so as to not leave out problems towards the end of the exam. • Write solutions on the exam sheet. If extra space is needed use the back of a page. Name: (print clearly) Signature: (for acceptance of honor code) Problem 1 (15 points) Problem 2 (15 points) Problem 3 (20 points) Problem 4 (15 points) Problem 5 (15 points) Problem 6 (10 points) Problem 7 (10 points) Total (100 points) Your TA/discussion section (circle one): Antebi (15, 18) Ayala (3, 6) Easton (14, 17) Fernanadez (2, 5) Kim (8, 11) Koytcheff (9, 12) Lo (21, 24) Rosales (26, 27) Tzeng (20, 23) Zamfir (29, 30) Schultz (51A) 1. (a) Find the reduced echelon form of 1 0 1 −1 0 0 1 2 −3 0 0 1 2 0 3 1 0 1 2 3 . (b) Consider the matrix A = 1 1 11 2 a 1 3 b where a and b are real numbers. Show that the null space of A is either {0} or a line, and give conditions on a and b that guarantee the null space of A is a line. 1 (3. continued) (c) Find a linear dependence relation between the columns {a1,a3,a4} of A, and explain your answer. (d) If b = 2a1 + 3a4 ∈ R3, where a1 and a4 are the first and fourth columns of A, find all solutions x of Ax = b. [Hint: What is one solution?] 4 4. (a) Assume that V and W are linear subspaces of Rn. Recall that vectors a, b ∈ Rn are orthogonal if a · b = 0. Let S be the set of all vectors w ∈ W that are orthogonal to every vector v ∈ V . Prove that S is a linear subspace of Rn. (b) Suppose L ⊂ R2 is a line through 0. Let x ∈ R2 be a vector not in L. Let H be the set of all vectors of the form v + tx, where v ∈ L and t ≥ 0 is a nonnegative scalar. Draw a picture of H. Is H a linear subspace of R2? If yes, prove it. If no, which of the defining conditions for a subspace hold for H and which fail? 5 5. (a) Let T : R2 → R2 be a linear transformation with T ([ 2 1 ]) = [ 2 −3 ] and T ([ 1 1 ]) = [ 4 1 ] . Find a matrix B so that Bx = T(x) for all x ∈ R2. [Hint: What is [ 2 1 ] − [ 1 1 ] ?] (b) Let S : R2 → R2 be the transformation that rotates vectors about the origin θ radians counterclockwise, where θ is the angle adjacent to the side of length 3 in a 3, 4, 5 right triangle. Find a matrix A so that Ax = S(x) for all x ∈ R2. 3 4 5 θ (c) With T and S as in parts (a) and (b), find a matrix C so that Cx = (S ◦T)(x) for all x ∈ R2. 6
