1 MAT208 Review for Exam II Fall 2009 For the given systems of equations, find as many as possible of the following items: 1. Express b as a linear combination of the columns of A. 2. Find bases for the column space, nullspace, row space and left nullspace of A. 3. Find the rank of A. 4. Find the determinant of A. 0 1 3 32 4 1 54 2 5 3Ab 1 2 1 1 42 1 3 1 61 7 6 2 6Ab 5. Are the following vectors linearly dependent or linearly independent? a. 1 2 3(1,0,2,1) v (0, 2,2,3) v (1, 2,4,4)v b. 1 2 3(1,1,8,1) v (1,0,3,0) v (3,1,14,1)v 6. Do the following vectors span the given space? a. 1 2 3(1,1,1) v (0,1,1) v (0,1,1)v in3? b. 1 2 3( 1,3,4) v (1,5, 1) v (1,13,2)v in3? 7. Do the vectors 1 2 3( 1,3,4) v (1,5, 1) v (1,13,2)v form a basis for3? What about1 2 3(1,3,4) v (2,7,2) v ( 1,2,1)v? 8. Is the set of all 2 x 2 matrices with the usual definition of scalar multiplication and addition defined as A + B = 0 (the zero matrix) a vector space? 9. If a 7 x 9 matrix has rank = 5, what are the dimensions of the four fundamental subspaces? 10. Find a complete solution ()pnx x x to the system 44x y zx y z.2 11. The set V of all 2 x 2 matrices is a vector space. Is the set of all symmetric 2 x 2 matrices a subspace of V? (A matrix is symmetric ifTAA). 12. If 1 2 3,,w w w are linearly independent vectors, show that the sums 1 3 2 1 3 3 1 2, , and zv w w v w w v w w are also linearly independent. 13. Which of the following are subspaces of 3? (a) all vectors of the form (a, 0, 0) (b) all vectors of the form (a, 1, 1) (c) all vectors of the form (a, b. c), where b = a + c (d) all vectors of the form (a, b, c), where b a + c + 1. 14. Let AB AC. Prove that if det( ) 0A, then B = C. 15. Find the determinant. 16. 17. Prove that 1det( )det( )det( )AABB. 18. Prove that a vector space has only one zero
View Full Document