Math 2090 Spring 2005 Sections 4 and 7 P. Achar Summary: Math 2090 Spring 2005 Sections 4 and 7 P. Achar Hour Exam 3 Solutions April 12, 2005 Total points: 100 Time limit: 50 minutes No calculators permitted. You must show all your work to receive full credit. IMPORTANT: You must answer Problems 1­5 and Problem 11. For Problems 6­10, choose three out of the five to answer. Note: in order to do Problem 11, you must answer at least one of Problems 9 or 10. 1. (4 points) What is the definition of dimension? Solution: the number of vectors in a basis for the vector space 2. (6 points) Let v1, v2, v3, v4 be vectors in a vector space V . In each of the following situations, what can you conclude about the dimension of V ? Your answers may be inequalities like "dim V 2," precise answers like "dim V = 5,", or the words "no information." (a) v1, v2, v3, v4 are linearly dependent. Solution: no information (b) v1, v2, v3 span V . Solution: dim V 3 (c) v1, v3, and v4 span V and are linearly independent. Solution: dim V = 3 3. (12 points) Consider the linear transformation T : P2 R, T(p(x)) = Collections: Mathematics