Linear Algebra Test #1 March 21, 2003 Name____________________ R.  Hammack Score ______

(1) (9 points) This problem concerns the vectors ,  ,  .
(a)

(b)

(c) Two of the vectors u, v, and w are orthogonal. Which two, and why?

(2) (10 points) Find the distance between the point in   and the plane .

(3) (10 points) Find the solution of the following linear system. Write the solutions in vector form.

(4)
(a) (10 points) Consider the set of vectors    in .  Is this a line or a plane?  If it is a line, write its equation in vector from. If it is a plane, write its equation in normal form.

(b) (10 points) Consider the set of vectors    in .  Is this a line or a plane?  If it is a line, write its equation in vector from. If it is a plane, write its equation in normal form.

(5) (10 points) Suppose A, B and X are invertible n-by-n matrices. Solve the following equation for X.

(6) This problem concerns the vectors , , .

(a) (10 points) Are these vectors linearly independent or linearly dependent? Show your work.

(b) (5 points) Is the span of these vectors equal to ?  Why or why not?

(7) (10 points) Suppose A is a 3-by-4 matrix. Are the columns of A linearly independent, linearly dependent, or is there not enough information to say?  Explain.

(8) (10 points) This problem concerns the invertible matrix
(a) Find .

(b) (6 points)  Use your answer to part (a) above to find a solution to the equation .