MATH 218 Exam 2, Fall 2014

Math 218 Quiz 2 Fall 2009 Name: Student ID: Instructions: 1. You must show correct work to receive credit. Correct answers with inconsistent work will not be given credit. 2. Books, notes and calculators are not allowed. 3. Turn off and put away all cell phones. Page Points Points Possible 2 12 3 12 4 14 5 10 6 12 Total 50 1 Math 218 Quiz 2 Fall 2009 Name: 1. (6 pts) Let V =      2a + b + 3c 2b + 2c 4a − 3b + c   ������ a, b, c ∈ R    . Find dim V . 2. (6 pts) Let T : R3 → R3 be a mapping with T     −1 2 0     =   3 0 a   , T     0 4 3     =   −1 b 5   , T     4 0 6     =   c 6 2   . For which values of a, b, c is T a linear operator? 2 Math 218 Quiz 2 Fall 2009 Name: 3. (6 pts ) Let T : V → W be a linear transformation and let U be a subspace of W. Prove that the set S = {v ∈ V |T(v) ∈ U} is a subspace of V . 4. (6 pts) Let T : V → V be a linear operator. Determine whether the set W = {v ∈ V |T(v) = v} is a subspace of V . 3 Math 218 Quiz 2 Fall 2009 Name: 5. (4 pts) Let T : R2 → R2 be given by T �� x y �� = � |x| |y| � where |x| is the absolute value of x. Determine whether T is a linear transformation. 6. Let T : R3 → R3 be given by T     x y z     =   x + 2y y − z x + 2z   . (a) (6 pts) Find a basis for N(T). (b) (4 pts) Find dim(R(T)). 4 Math 218 Quiz 2 Fall 2009 Name: 7. (4 pts) Let {v1, v2, . . . , vk} be a linearly independent set of vectors in a vector space V . Show that the set {v1, v2, . . . , vk−1} cannot be a basis for V . 8. (6 pts) Let V = �� 1 1 + t 2 + t 2 ����� t ∈ R � . Define � 1 1 + t 2 + t 2 � ⊕ � 1 1 + s 2 + s 2 � = � 1 1 + t + s 2 + t + s 2 � . Find the additive identity and additive inverse. 5 Math 218 Quiz 2 Fall 2009 Name: 9. (4 pts each) Determine whether each of the following statements is true or false. Justify your answer. (a) If {a, b, c} is linearly dependent then a is a linear combination of b and c. (b) If T : R12 → R7 is a linear transformation, then dim(N(T)) ≥ 5. (c) If W is a subspace of a vector space V , then V − W, the set of all vectors that are in V but not in W, is also a subspace of V . 6