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

By the addition to Theorem 2.7 (p. 98) given in class, the rows of matrix A with m rows and n columns spanR^{n}if and only if rank(A) = n. So we can do the part about the spans just knowing the rank. Moreover Theorem 2.7 says the rows are linearly dependent if and only if m > rank(A), so we can do the part about linear independence also just knowing the rank. To save myself some typing, I won't show my work row reducing the matrices, but you should show it! (a) A= [1 4 0 4 -1] [0 0 1 4 -1] [1 4 1 8 -2] We immediately see that the rows do not spanR^{5}, since the rank of a matrix with m rows and n columns is at most the minimum of m and n, so here rank(A) <= 3. To check independence of the rows we need to row reduce. R_{A}= [1 4 0 4 -1] [0 0 1 4 -1] [0 0 0 0 0] The rank is 2, so the rows are linearly dependent (in fact, the bottom row is the sum of the top two rows). (b) A= [1 1 1] [0 1 -1] [0 1 -1] R_{A}= [1 0 0] [0 1 0] [0 0 0] The rank is again 2, so the rows are linearly dependent but do not spanR^{3}. (c) A= [1 2 3] [1 0 4] [0 2 0] R_{A}= [1 0 0] [0 1 0] [0 0 1] The rank is 3, so the rows are linearly independent and spanR^{3}.

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? Answer: the free variables are the ones
corresponding to columns without a leading 1; i.e., x
_{2}and x_{5}. - (b) How many solutions does the system of equations have? Answer: it is a consistent since the augmented column does not have a leading 1, so there is at least one solution, and it has free variables, so there are infinitely many solutions if there are any (which there are because it is consistent).
- (c) Write down the general solution of the system of equations
in vector form.
[x

_{1}] [-2] [6] [-3] [x_{2}] [ 0] [1] [ 0] [x_{3}] = [ 7] + x_{2}[0] + x_{5}[-4] [x_{4}] [ 8] [0] [-5] [x_{5}] [ 0] [0] [ 1]

- (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. Answer: True. (There must be a free variable, so not every variable can be a leading variable; i.e., there are more variables than leading variables. But the number of leading variables is the same as the number of leading 1's in the reduced row echelon form of A, which is the rank of A.)
- (b) A linear system with fewer variables than equations never has infinitely many solutions. Answer: False. (For example, x+y=0, 2x+2y=0 and 3x+3y=0 has 2 variables, 3 equations and infinitely many solutions.)
- (c) A homogeneous linear system with a unique solution must have at least as many equations as variables. Answer: True. (It can't have a free variable if it has only one solution, so each variable must be a leading variable, so the rank of the coefficient matrix A equals the number of variables. But the number of rows of A (which is also the number of equations) is always at least equal to the rank. So there are at least as many equations as variables.)
- (d) A linear system with fewer equations than variables is never inconsistent. Answer: False. (For example, x+y+z=0, 2x+2y+2z=1 is inconsistent.)
- (e) Suppose a matrix has n columns, and that
the matrix is taller than it is wide.
Then the rows span
**R**^{n}. Answer: False. (The matrix could, for example, be the 0 matrix.) - (f) Suppose a matrix is taller than it is wide. Then the rows of the matrix are linearly dependent. Answer: True. (This is what Theorem 2.8 says.)

(b) Think of u and v as adjacent sides of a parallelogram as shown in the figure. Then u+v and u-v are the diagonals. Thus (a) says that the diagonals of a parallelogram have the same length if and only if the adjacent sides are perpendicular (i.e., if and only if the parallelogram is a rectangle).

- (a) Find the cosine of the acute angle between the
planes defined by x+y+z=2 and x+2y+z=0.
Answer: Just find the absolute value of the
cosine of the angle between the normal vectors
for the planes: (1,1,1).(1,2,3)/[||(1,1,1)||*||(1,2,3)||]
= 6/(3*14)
^{1/2}. - (b) Find an equation (in vector form)
for the line L where the two planes intersect.
Answer: Find any two solutions P and Q to the system of equations given by the
equations of the two planes. For example, P = (1, -2, 3) and Q=(4, -2, 0).
Then the vector
**d**= (3, 0, -3) from P to Q is a direction vector for L, so**x**= P + t**d**gives an equation for the line L. - (c) Find (in normal form) an equation for the
plane passing through the point R=(7,3,2) and perpendicular to L.
Answer: The vector
**d**is a normal for the desired plane, so the normal form is**d**.(**x**-R)=0, where**d**and R are as above.