advertisement

Math 20F Test 2 100 points February 25, 2009 --------------------------------------------------------------------------------------------------Notation: All scalars and matrix entries and polynomial coefficents are in the set R of real numbers. Points: Problems (1), (3), (5), (6ii) 18 pts each ; (2), (4) 10 pts each ; (6i) 8 pts Directions: Show all work and justify all answers. ----------------------------------------------------------------------------------------------------(1) Find a basis for the column space CS(A), where A denotes the 4 by 4 matrix 1 2 4 6 2 4 3 2 3 6 4 2 4 8 8 8 ------------------------------------------------------------------------------------------------------(2) True or false: The determinant of the 4 by 4 matrix A in problem (1) equals zero. ------------------------------------------------------------------------------------------------------(3) Let A be a particular 3 by 3 matrix whose column space CS(A), viewed geometrically, is a line through the origin. True or False: The null space NS(A) of this matrix A, viewed geometrically, must be a plane through the origin. -------------------------------------------------------------------------------------------------------(4) Let V be the vector space over R consisting of all polynomials in x whose degree is at most 7 (including all constant polynomials such as 0, 1, etc.). For example, the polynomial x5 - 2x3 + x + 4 is in V because it has degree 5, but the polynomial x8 - 2x3 + x + 4 is not in V, because it has degree 8. Find the dimension of the vector space V. ---------------------------------------------------------------------------------------------------------(5) Consider the 3 by 2 matrix A and the 3 by 1 vector b defined by 2 1 b1 A = -4 3 , b = b2 . -6 4 b3 Show that b is in the column space CS(A) if and only if b3 = (7 b2 - b1 ) / 5 . -----------------------------------------------------------------------------------------------------------(6) Suppose that the 2 by 2 matrix 0 1 2 0 represents a linear transformation T from R2 to R2 with respect to a certain basis v1 , v2 for R2 . (i) True or false: T(v2) = v1 . (ii) True or false: The 2 by 2 matrix 0 2 1 0 represents the transformation T above with respect to the new basis v1 + v2 , v1 + 2v2 .