Math 3130 - Intro to Linear Algebra - MathOnline
Department of Mathematics - University of Colorado at Colorado Springs
Instructor: Dr. Radu C. Cascaval
Phone: 719-255-3759, Fax: 1-888-951-4321, Email: radu@uccs.edu
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March 8-9, 2016
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Exam 1 - Math 3130 - Intro to Linear Algebra
Dr. Radu C. Cascaval, Spring 2016
The exam has 8 problems on 5 pages. Make sure you turn in ALL pages. To receive credit, show all work.
No calculators with linear algebra capabilities allowed!
1. In each case the augmented matrix for a system of linear equations has been reduced by row operations to the
given form. For each matrix answer the following:
(a) is the matrix in reduced row echelon form, row echelon form, or neither,
(b) is the linear system consistent or inconsistent,
(c) find the general solution, if applicable.


1 0 2 1
(i)  0 1 3 0 
0 0 0 1

1
(ii)  0
0

2
0
0

1
0 
0
1 1 0
(iii)  0 1 1
0 1 0

0
0 
1
2. Given the following matrices

4
A= 1
2

−1
1 ,
3
B=
0
1
1
2
compute the following matrices, if possible, otherwise explain why it is impossible.
AB,
BA,
(AB)T ,
AT B T ,
A−1
3. Using the inversion algorithm, decide whether the matrix below is invertible or not. If it is, find its inverse.


1 3 1
A= 2 2 1 
2 3 1
4. Determine the LU decomposition of the matrix

1
A= 3
1
2
6
2

−1
−1  .
−3
Show all your work. Check your answer at the end.
5. Determine whether there exist scalars c1 , c2 , c3 such that
 
 
   
5
1
2
3
c1 2 + c2 5 + c3 3 =  3 
17
1
0
8
or not. If yes, find them.
 
 
1
1
0
1



in R4 , find two vectors v3 and v4 that are orthogonal to both v1
6. Given the vectors v1 =   and v2 =  
1
5
4
4
and v2 . [Make sure v3 is not a scalar multiple of v4 ]
7. Verify that the Cauchy-Schwarz inequality holds for the following pair of vectors
(a) ~u = (4, 1, 1), ~v = (1, 2, 3)
(b) ~u = (1, 2, 1, 2, 3), ~v = (0, 1, 1, 5, −2)
In each case, find the cosine of the angle θ between these vectors.
8. Find the general solution of the system Ax = b with the augmented

1 2 0 0 0
0 0 1 3 0
rref ([A|b]) = 
0 0 0 0 1
0 0 0 0 0
matrix in reduced row echelon form

2
4

−1
0