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Name: ID#: Name: ID#: 8. Give complete answers. 14. Determine which of the following sets are linearly independent. Give reasons for your a. Describe all of the subspaces of R2. answers . b. Describe all of the subspaces of R3. 9. Suppose A is an m x n matrix. a. What is the null space of A ? b. What is the column space of A ? c. Let b E Rm. Explain how to determine whether b E Col A. (2x - y+z=-1 10. Find the general solution to the system ( -x + 2y + z = 2 xty+ 2z=1. d. (1+x, 2x-x', x - x-1,x2 + 2x) 11. 4 =6 ; ) a. Give the value(s) of x for which A is singular. b. Give the value(s) of x for which A is invertible. 15. The elementary row operations R1 R2, -2R1 + R3 - R3, - R2 - R2 were applied (in the order given) to the matrix A to obtain the matrix a. Find the solution to Ax = b without finding the matrix A. a. A = -1 2 2 b. ATA = b. Find the solution to Ax = 1 0 1 6-x without finding the matrix A. 16. The matrix linear transformation T(u) = (_, )u transforms the triangle with vertices -1 1 3 (-1,3), (-2,5) and (3,7) into a new triangle. Use the area of the old triangle and a 13. A = determinant to find the area of the new triangle a. Give a basis for the null space of A. 17. Suppose A is an invertible n x n matrix, but you don't have A. You only have A-1. b. Give the null space of AT. Explain how to use EROs to find the matrix A. 18. What must be true for a subset U of R" to be a subspace of