Math 317: Linear Algebra
Practice Exam 3
Fall 2015
Name:
*Please be aware that this practice test is longer than the test you will see on December
4, 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.
 
3
1
4

1. Find the closest point to y = 
5 in the subspace of R spanned by the vectors
1
v1 = (3, 1, −1, 1) and v2 = (1, −1, 1, −1).
2. Let W be the subspace of R3 given by W = {(x1 , x2 , x3 ) : x1 + 2x2 + 3x3 = 0}.
(a) Find an orthonormal basis {b1 , b2 } for W .
(b) Find a third unit vector b3 that is orthogonal to both b1 and b2 so that
B = {b1 , b2 , b3 } is an orthonormal basis for R3 .
(c) Let T : R3 → R3 be the linear transformation that reflects x ∈ R3 across W .
Write the matrix of T with respect to B.
(d) Write the matrix of T with respect to the standard basis.
3. Let

0
0
C=
1
0
1
0
0
0
0
1
0
0

0
0
,
0
2

1
0
D=
0
0

3 5 −9
2 −2 1 
.
0 1 −2
0 0
2
Compute det C, det D and det CD.
4. (a) Let E be an elementary matrix that corresponds to one of the three types of
elementary row operations. Prove that det E = det E T .
(b) Prove that for any square matrix A, det A = det AT .
(c) Prove that if A has a column of 0 entries then det A = 0 without directly using
the fact that A is singular.
5. Let A and B be 4 × 4 matrices and suppose that det A = 5 and det B = 3. Compute
the following if possible.
(a) det AT
(b) det A−1
(c) det AB
(d) det(A + B)
1
Math 317: Linear Algebra
Practice Exam 3
Fall 2015
(e) det(2A)
(f) det(−B)
6. Determine whether {t, t + 1, t + 2} forms a linearly independent subset of P1 , the
space of all polynomials of degree 1 or less.
7. Find a QR factorization for A and use this QR factorization to solve Ax = b where


2 1
1 1 

A=
0 1  ,
1 −1


2
1

b=
1
−1
2