Name:
AMAT 220: Linear Alge
a
Practice Exam
March, 2021
Show all work for each problem in the space provided. If you run out of room for an answer, continue on
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Question Points Bonus Points Score
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2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
9 0 0
Total: 0 0
1. Define what both a linear transformation is and the span of a set of vectors.
2. Find the general solution of the linear system
3x1 − 4x2 + 2x3 = 0 (1)
−9x1 + 12x2 − 6x3 = 0 (2)
−6x1 + 8x2 − 4x3 = 0 (3)
OR
x1 − 7x2 + 6x4 = 5 (4)
x3 − 2x4 = −3 (5)
−x1 + 7x2 − 4x3 + 2x4 = 7 (6)
y row reducing the co
esponding augmented matrix and interpreting your result in terms of the co
esponding
linear system.
3. Show by computation whether the given vector b ∈ Span{a1, a2, . . . , an}, where ai are vectors. Specifically,
show whethe
=

−9
−7
−15

is contained in the spanning set
Span{a1 =

2
1
4
 a2 =

1
−1
3
 a3 =

3
2
5
}
OR
=

9
2
7

is contained in the spanning set
Span{a1 =

1
2
−1
 a2 =

6
4
2
}
4. Describe in parametric form all solutions of the system Ax = 0 for the given A.
A =
2 1 3
1 2 0

OR
A =
3 1 1 1
5 −1 1 1

5. Determine if the given set of vectors are linearly dependent:
S = {a1 =

−2
0
1
 , a2 =

3
2
5
 , a3 =

6
−1
1
 , a4 =

7
0
2
}
OR
S = {a1 =

8
−1
3
 , a2 =

4
0
1
}
6. Compute the standard matrix A associated to the given linear transformation T : Rn → Rm, where
T (x1, x2) = (x1 + 2x2, 3x1 − x2)
OR
T (x1, x2, x3) = (2x1 − x2 + x3, x2 − 4x3)
7. Compute the matrix product AB for the given A and B.
A =
1 2 4
2 6 0

and
B =

4 1 4 3
0 −1 3 1
2 7 5 2

OR
A =

6 1 3
−1 1 2
4 1 3

and
B =

3 0
−1 2
1 1

8. Compute the inverse of the given matrix A by the algorithm for computing a matrix’s inverse; that is, row
educe the matrix of A augmented by I.
A =

1 2 3
2 5 3
1 0 8

OR
A =

3 4 −1
1 0 3
2 5 −4

9. Use your answer from the question 8. to find a solution x to the linear system
x1 + 2x2 + 3x3 = 7 (7)
2x1 + 5x2 + 3x3 = 5 (8)
x1 + 8x3 = 8 (9)
RESPECTIVELY
3x1 + 4x2 − x3 = 7 (10)
x1 + 3x3 = 5 (11)
2x1 + 5x2 − 4x3 = 8 (12)
Respectively means apply the inverse of the first answer in 8 to solve the first problem in this question and to
apply the second answer in 8 to solve the second problem in this question.