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just answer all quesitons and show works just answer all quesitons and show worksjust answer all quesitons and show worksjust answer all quesitons and show worksjust answer all quesitons and show worksjust answer all quesitons and show worksjust answer all quesitons and show works
just answer all quesitons and show works
Name:
AMAT 220: Linear Algebra
Practice Exam
March, 2020
Show all work for each problem in the space provided. If you run out of room for an answer, continue on
the back of the page. You may NOT use a calculator
Question
Points
Bonus Points
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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
3×1 − 4×2 + 2×3 = 0
(1)
−9×1 + 12×2 − 6×3 = 0
(2)
−6×1 + 8×2 − 4×3 = 0
(3)
OR
x1 − 7×2 + 6×4 = 5
x3 − 2×4 = −3
−x1 + 7×2 − 4×3 + 2×4 = 7
(4)
(5)
(6)
by row reducing the corresponding augmented matrix and interpreting your result in terms of the corresponding
linear system.
3. Show by computation whether the given vector b ∈ Span{a1 , a2 , . . . , an }, where ai are vectors. Specifically,
show whether
−9
b = −7
−15
is contained in the spanning set
2
1
3
Span{a1 = 1 a2 = −1 a3 = 2}
4
3
5
OR
9
b = 2
7
is contained in the spanning set
1
6
Span{a1 = 2 a2 = 4}
−1
2
4. Describe in parametric form all solutions of the system Ax = 0 for the given A.
2 1 3
A=
1 2 0
OR
3
A=
1
1 1
5 −1 1 1
5. Determine if the given set of vectors are linearly dependent:
−2
3
6
7
S = {a1 = 0 , a2 = 2 , a3 = −1 , a4 = 0}
1
5
1
2
OR
8
4
S = {a1 = −1 , a2 = 0}
3
1
6. Compute the standard matrix A associated to the given linear transformation T : Rn → Rm , where
T (x1 , x2 ) = (x1 + 2×2 , 3×1 − x2 )
OR
T (x1 , x2 , x3 ) = (2×1 − x2 + x3 , x2 − 4×3 )
7. Compute the matrix product AB for the given A and B.
1 2 4
A=
2 6 0
and
4 1 4 3
B = 0 −1 3 1
2 7 5 2
OR
6
1 3
A = −1 1 2
4 1 3
and
3
0
B = −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
reduce the matrix of A augmented by I.
1 2 3
A = 2 5 3
1 0 8
OR
3 4 −1
A = 1 0 3
2 5 −4
9. Use your answer from the question 8. to find a solution x to the linear system
x1 + 2×2 + 3×3 = 7
(7)
2×1 + 5×2 + 3×3 = 5
(8)
x1 + 8×3 = 8
(9)
3×1 + 4×2 − x3 = 7
(10)
x1 + 3×3 = 5
(11)
2×1 + 5×2 − 4×3 = 8
(12)
RESPECTIVELY
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.
