Math 317: Linear Algebra
Practice Exam 1
Fall 2015
Name:
*Please be aware that this practice test is longer than the test you will see on September
28, 2015. Also, this test does not cover every possible topic that you are responsible for
on the exam. For a comprehensive list of all topics covered on the exam, please see the
exam topics document on the website.
I have purposefully strayed away from computation type problems (except for one or two)
to give you some additional insight and practice on the types of proofs you could see on
the exam. The exam is roughly 40% theory and 60% computational, so please be sure
that you know how to compute answers as well! (See Exam Guide topics).
1. Suppose that a, b, c ∈ Rn such that a is parallel to b and b is parallel to c. Prove
that span(a, b, c) = span(a).
2. Find a parametric equation for the line through points A = (1, 2, 1) and B = (2, 1, 0).
3. Determine if the following property is true or false. If it is true, prove it. If it false,
provide a counterexample or explain why the statement cannot be true.
For any vectors x, y, z ∈ Rn , x · (y · z) = (x · y) · z.
4. Prove that for any vectors x, y ∈ Rn , we have that
1
kx + yk2 + kx − yk2 = kxk2 + kyk2 .
2
5. Does there exist a system of linear equations with exactly 3 solutions? If yes, provide
an example; if not, explain why not.
6. Suppose that A is an n×n matrix that is both upper triangular and lower triangular.
Prove that A is a diagonal matrix.
7. If A2 = B, prove that both A and B must be square matrices.
8. Explain why the following claim is false:
If A is a 2 × 2 matrix such that A2 = 0 where 0 denotes the zero matrix, then A is
nonsingular.
9. Prove that if T is a linear transformation from Rn to Rn , then T (0) = 0.
1