Instructions: Show all of your work; your work must justify your answers.

Note: These are mostly questions I've given on exams when I've taught M314 before, so this should give you an idea what my exams are like. And although it should help you to review, this is not a review sheet in the usual sense. There may be things you need to know that aren't on it, so be sure to review the book, the homework and the quizzes. On the other hand, problem 5 will be on the exam Wednesday (verbatim!), so be sure you know how to do problem 5.

Problem 1 (18 points): For each matrix A below, find its reduced row echelon form R

(a) A= [1 4 0 4 -1] [0 0 1 4 -1] [1 4 1 8 -2] (b) A= [1 1 1] [0 1 -1] [0 1 -1] (c) A= [1 2 3] [1 0 4] [0 2 0]

Problem 2 (15 points): Suppose

[1 -6 0 0 3 | -2] [0 0 1 0 4 | 7] [0 0 0 1 5 | 8] [0 0 0 0 0 | 0]is the reduced echelon form of the augmented matrix of a system of 4 equations in the variables x

- (a) Which variables are the free variables?
- (b) How many solutions does the system of equations have?
- (c) Write down the general solution of the system of equations in vector form.

- (a) If A is the coefficient matrix of a linear system with infinitely many solutions, then the number of variables is more than the rank of A.
- (b) A linear system with fewer variables than equations never has infinitely many solutions.
- (c) A homogeneous linear system with a unique solution must have at least as many equations as variables.
- (d) A linear system with fewer equations than variables is never inconsistent.
- (e) Suppose a matrix has n columns, and that
the matrix is taller than it is wide.
Then the rows span
**R**^{n}. - (f) Suppose a matrix is taller than it is wide. Then the rows of the matrix are linearly dependent.

- (a) Find the cosine of the acute angle between the planes defined by x+y+z=2 and x+2y+z=0.
- (b) Find an equation (in vector form) for the line L where the two planes intersect.
- (c) Find (in normal form) an equation for the plane passing through the point (7,3,2) and perpendicular to L.