Author: Anonymous

This test is not worth 100 points.  However, your percent gr…

This test is not worth 100 points.  However, your percent grade will be recorded.  Show as much work as you can and provide the appropriate justifications.  You may use technology to complete row reductions and find inverses as needed.  The problems that indicate an explanation is needed, you could just provide supporting work as your justification.    1. 12pts.  Assume that the matrix A is row equivalent to B below. Find a basis for Col A, Row A, and Nul A.   2. 6pts. Determine the rank and nuillity of . 3.  8pts. Find the basis for the given subspace and state the dimension:   4. Basis a. 7pts. Verify that {1, 1-t, 2 – 4t + t2} is a basis for (all second degree polynomials) by checking the definition. b. 3pts. Find the coordinates of p(t)= 7 – 8t +3t2 with respect to the basis.   5. 10pts.  Let H be the set of all polynomials of the form p(t) = a + bt2 where a and b are in and b>a. Determine whether H is a vector space. If it is not a vector space, determine which of the following properties that it fails to satisfy. If there is more than one property it fails, identify all properties that fail.  Provide as much supporting work as possible or counterexamples.  A: Contains a zero vector. B: Closed under addition. C. Closed under multiplication of scalars.  6. 6pts. Find the vector x given the basis B and ;  . 7. 6pts. Find the coordinate vector for the given x and basis B:  . 8. 5pts. If A is a 6 x 4 matrix, what is the smallest possible dimension of Nul A? Explain. 9. 5pts. If A is a 4 x 3 matrix, what is the largest possible dimension of the row space of A?  If A is a 3 x 4 matrix, what is the largest possible dimension of the row space of A? Explain. 10. 6pts. If the null space of a 5 x 7 matrix is 3-dimensional, find Rank A and dim Row AT. Explain. 11. 6pts. A scientist solves a nonhomogeneous  system of ten linear equations in twelve unknowns and finds that three of the unknowns are free variables. Can the scientist be certain that, if the right sides of the equations are changed, the new nonhomogeneous system will have a solution? Explain. 12. 5pts. Is it possible that all solutions of a homogenous system of nine linear equations and twelve unknowns are multiples of one fixed nonzero solution? Explain.