M314 Practice Exam 1
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 RA
and determine its rank. Also, in each case, determine
if the rows of the given matrix span Rn,
where n in each case is the number of columns of the matrix,
and determine if the rows
of the given matrix are linearly independent.
Show your work and explain your answers.
(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 x1,..., x5.
Problem 3 (12 points): For each of the following statements,
determine whether it is True or False; if it is false,
give a specific example which shows it is not true
(it is not enough to just say in general terms why it is false).
- (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.
Problem 4 (15 points): Problem 21 p. 100. (On the actual exam,
I will type out every problem.)
Problem 5 (15 points) Do Problem 58 on p. 27.
Use complete sentences and good grammar in your answer!
Problem 6 (10 points) In Problem 14 on p. 14, write the vector
from C to E as a linear combination of a and b.
Problem 7 (15 points) Consider the planes defined by
x+y+z=2 and x+2y+z=0.
- (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
- (e) Suppose a matrix has n columns, and that
the matrix is taller than it is wide.
Then the rows span Rn.
- (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.