Practice Final Exam - Math 3130 - Intro to Linear Algebra
Dr. Radu C. Cascaval - Spring 2016
Below is a sampple final exam, and does not cover all topics that may appear on the actual final exam. Please review
all sections covered in class, the homework assignments and other similar exercises at the end of each covered section.
Also supplementary exercises at the end of each covered chapter are a great sources of practice problems.
Show all work. No calculators with linear algebra capabilities allowed during exams!
Exercise 1.[6 pts each]
Determine whether each of the following statements are TRUE or FALSE.
Circle your answer and justify it for credit. No justification, no credit.
• A matrix A is invertible if λ = 0 is not an eigenvalue for A.
TRUE
FALSE
• If two n × n matrices A and B have the same row space, then they also have the same column space.
TRUE
FALSE
• If the system Ax = b has at least one solution, then b is in the column space of A.
TRUE
FALSE
• An orthogonal matrix Q (by definition QT Q = I) must be square.
TRUE
FALSE
• Any subspace W of Rn admits an orthonormal basis.
TRUE
FALSE
• If Ax = b is an inconsistent m × n system, then AT Ax = AT b is always consistent.
TRUE
FALSE
Exercise 2.
[8 pts each]
1 2
Consider the matrix A = −1 1
1 5
(a) Using Gauss-Jordan, show that

0
4.
4
the matrix A is not invertible;
(b) Find the LU decomposition of A.
(c) Solve the homogeneous system Ax = 0.
(d) Determine the column space of A, null space of A, row space of A and the left nullspace of A.
Exercise 3. [10 pts each]
(a) Determine whether the vectors ~v1 = [−2, 1, 0]T , ~v2 = [−2, 1, −1]T and ~v3 = [1, 0, 2]T are linearly independent
vectors in R3 ,
(b) Express w
~ 1 = [1, 1, 1]T as a linear combination of the three vectors ~v1 , ~v2 , ~v3 .
Exercise 4. [12 pts each]
Using Cramer’s Rule, solve the system


 x1 + x2 + x3
2x1 − x2


4x1
− x3
=1
=0
=0
Exercise 5.
[16 pts each]

1
−1
(a) Using Gram-Schmidt, find an orthonormal basis for the column space W of the matrix A = 
0
0
(b) Find W ⊥ , the orthogonal complement of the column space.
0
1
−1
0

0
0

1
−1
Exercise 6.
[20 pts]
Find the best approximation solution (in the least

2
A = 1
3
square sense) to the to the linear system Ax = b, where

 
−2 0
2
1 2 , b = −1 .
1 4
1
Exercise 7. [ 10 pts each]
Given the matrix,

1
(i) A = −4
2
(a) Find all eigenvalues and eigenvectors of the matrix A.
−4
1
−2

2
−2 .,
−2
(b) Determine whether A is diagonalizable or not. If yes, find a diagonal matrix D and an invertible P such that
P −1 AP = D.
(c) Determine whether A is orthogonally diagonalizable or not. If yes, find a diagonal matrix D and an orthogonal
matrix Q such that QT AQ = D.
Exercise 8. [ 18 pts]
Redo exercise 7 in its entirety (a), (b), (c) for the matrix

0 0
A = 1 2
1 0

−2
1
3