MATH 218 Exam 2, Fall 2010

Math 218 Quiz 2 Fall 2010 Name: Student ID: Instructions: 1. This exam has 5 pages. Please make sure you have all pages. 2. The point value of each problem occurs to the left of the problem. 3. You must show correct work to receive credit. Correct answers with inconsistent work or with no justification will not be given credit. 4. Only non-graphing and non-programmable calculators are allowed. 5. Turn off and put away all cell phones. Page Points Points Possible 2 11 3 12 4 9 5 8 Total 40 1 Math 218 Quiz 2 Fall 2010 Name: 1. (4 pts) For which values of k do the vectors   −1 1 k   ,   0 1 3   and   k 2 2   form a basis for R3? 2. Let t : R3 → R3 be given by T     x y z     =   6x + 3y 4x + 2y + z 4z  . (a) (4 pts) Find a basis for R(T). (b) (3 pts) Is T an isomorphism? 2 Math 218 Quiz 2 Fall 2010 Name: 3. Let T : P2 → P1 be given by T(p(x)) = xp(1) + p(0). (a) (4 pts) Show that T is a linear transformation. (b) (4 pts) Find ker(T). (c) (4 pts) Find R(T). 3 Math 218 Quiz 2 Fall 2010 Name: 4. (4 pts) Let T : V → W be a linear transformation. Determine whether {v ∈ V | T(v) ̸= 0} is a subspace of V . 5. (5 pts) Suppose S and T are subspaces of a vector space V . Define W = {2u + 3v | u ∈ S, v ∈ T}. Determine whether W is a subspace of V . 4 Math 218 Quiz 2 Fall 2010 Name: 6. (3 pts) Find a linear transformation T : R3 → R3 and a linearly independent subset {v1, v2, v3} of R3 such that {T(v1), T(v2), T(v3)} is linearly dependent. 7. (5 pts) Let T : V → W be a linear transformation and let {v1, v2, v3} be a subset of V . Show that if T is one-to-one and {T(v1), T(v2), T(v3)} is linearly dependent then {v1, v2, v3} is also linearly dependent. 5