Linear Algebra
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test1review.docx
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1
Solve the system of equations.
1) x1 + x2 + x3 = 7
x1 - x2 + 2x3 = 7
5x1 + x2 + x3 = 11
A) (4, 1, 2) B) (1, 2, 4) C) (4, 2, 1) D) (1, 4, 2)
1)
Determine whether the system is consistent.
2) x1 + x2 + x3 = 7
x1 - x2 + 2x3 = 7
5x1 + x2 + x3 = 11
A) No B) Yes
2)
Determine whether the matrix is in echelon form, reduced echelon form, or neither.
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1 |
6 |
5 |
-7 |
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3) |
0 |
1 |
-4 |
-1 |
3) |
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0 |
0 |
1 |
3 |
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A) Echelon form B) Neither C) Reduced echelon form
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1 |
4 |
5 |
-7 |
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4) |
4 |
1 |
-4 |
7 |
4) |
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0 |
4 |
1 |
4 |
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A) Echelon form B) Reduced echelon form C) Neither
Find the indicated vector.
5) Let u = -2
-3
A)
18
-27
. Find -9u.
B)
-18
27
C)
-18
-27
5)
D)
18
27
6) Let u = -3
2
A)
21
-14
. Find 7u.
B) C)
-21 21
14 14
D)
-21
-14
6)
Compute the product or state that it is undefined.
7) [-6 2 5]
A)
[-51]
Write the system as a vector equation or matrix equation as indicated.
8) Write the following system as a vector equation involving a linear combination of vectors.
3x1 - 5x2 - x3 = 2
5x1 + 3x3 = 6
8)
A) x1 3
5
+ x2 -5
1
+ x3 1 = 2
3 6
x1
B) 3 x2 x3
3
x1
- 5 x2 -
x3
5
x1 2 x2 = 6 x3 0
2
|
C) x1 |
3 + x2 |
-5 + x3 |
-1 = |
2 D) x1 |
5 |
+ x2 |
0 |
= |
6 |
|
5 |
0 |
3 |
6 |
|
-1 |
|
3 |
|
0 |
Solve the problem.
9) Find the general solution of the homogeneous system below. Give your answer as a vector.
x1 + 2x2 - 3x3 = 0
4x1 + 7x2 - 9x3 = 0
-x1 - 3x2 + 6x3 = 0
9)
A)
x1 -3 x2 = x3 3 x3 0
C)
x1 3
B)
x1 -3 x2 = x3 3 x3 1
D)
x1 -3
x2 = x3 -3 x3 1
x2 = 3
x3 1
10) Find the general solution of the simple homogeneous ʺsystemʺ below, which consists of a single linear equation. Give your answer as a linear combination of vectors. Let x2 and x3 be free variables.
-2x1 - 14x2 + 8x3 = 0
A)
10)
x1 -7 x2 = x2 0 x3 1
B)
x1 x1
4
+ x3 1
0
x1
(with x2, x3 free)
x2 = -7 x2 x3 x3
C)
x1 7
- 4 x2 x3
-4
(with x2, x3 free)
x2 = x2
x3
D)
x1
x2 = x2
x3
1 + x3 0
0 1
-7 4
1 + x3 0
0 1
(with x2, x3 free)
(with x2, x3 free)
11) Suppose an economy consists of three sectors: Energy (E), Manufacturing (M), and Agriculture (A).
Sector E sells 70% of its output to M and 30% to A.
Sector M sells 30% of its output to E, 50% to A, and retains the rest. Sector A sells 15% of its output to E, 30% to M, and retains the rest.
Denote the prices (dollar values) of the total annual outputs of the Energy, Manufacturing, and Agriculture sectors by pe, pm, and pa, respectively. If possible, find equilibrium prices that make each sectorʹs income match its expenditures.
Find the general solution as a vector, with pa free.
11)
A)
pe
pm =
pa
C)
pe
pm =
pa
0.308 pa
0.716 pa pa
0.607 pa
0.481 pa
pa
B)
pe
pm =
pa
D)
pe
pm =
pa
0.465 pa
0.593 pa pa
0.356 pa
0.686 pa
pa
12) The network in the figure shows the traffic flow (in vehicles per hour) over several one -way streets in the downtown area of a certain city during a typical lunch time. Determine the general flow pattern for the network.
In other words, find the general solution of the system of equations that describes the flow. In your general solution let x4 be free.
13) For what values of h are the given vectors linearly independent?
12)
13)
1 -4
-6 , 24
1 h
A) Vectors are linearly independent for h = -4
B) Vectors are linearly dependent for all h
C) Vectors are linearly independent for all h
D) Vectors are linearly independent for h ≠ -4
1
14) Let v1 = -3
8
, v2 =
-3
8 , v3 =
5
2
-2 .
-6
14)
Determine if the set {v1, v2, v3} is linearly independent.
A) Yes B) No
Describe geometrically the effect of the transformation T.
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0 |
0 |
0 |
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15) Let A = |
0 |
1 |
0 |
. |
15) |
|
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|
0 |
0 |
1 |
|
|
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Define a transformation T by T(x) = Ax.
A) Projection onto the x2-axis B) Horizontal shear
C) Projection onto the x2x3-plane D) Vertical shear
Solve the problem.
16) Let T: ℛ2 -> ℛ2 be a linear transformation that maps u = -3
4
into -13
6
and maps v = 4
6
into
16)
|
6 |
. |
|
-8 |
|
Use the fact that T is linear to find the image of 3u + v.
A) B)
-5 -7
18 -2
C)
-21
-6
D)
-33
10
Determine whether the linear transformation T is one -to-one and whether it maps as specified.
17) T(x1, x2, x3) = (-2x2 - 2x3, -2x1 + 8x2 + 4x3, -x1 - 2x3, 4x2 + 4x3)
Determine whether the linear transformation T is one -to-one and whether it maps ℛ3 onto ℛ4.
A) One-to-one; onto ℛ4 B) One-to-one; not onto ℛ4
C) Not one-to-one; not onto ℛ4 D) Not one-to-one; onto ℛ4
17)
18) Let T be the linear transformation whose standard matrix is
18)
|
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1 |
-2 |
3 |
|
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A = |
-1 |
3 |
-4 |
. |
|
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2 |
-2 |
-9 |
|
Determine whether the linear transformation T is one -to-one and whether it maps ℛ3 onto ℛ3.
A) Not one-to-one; not onto ℛ3 B) One-to-one; not onto ℛ3
C) Not one-to-one; onto ℛ3 D) One-to-one; onto ℛ3
Find the matrix product AB, if it is defined.
19) A = 0 -2
2 3
, B = -1 3 2 .
0 -3 1
19)
A)
0 -2 6
-3 -2 7
B)
0 6 -2
-2 -3 7
C) AB is undefined. D)
0 -6 -12
0 -9 3
The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are defined.
20) A is 2 × 1, B is 1 × 1.
A) AB is 1 × 2, BA is 1 × 1. B) AB is 2 × 1, BA is undefined.
C) AB is undefined, BA is 1× 2. D) AB is 2 × 2, BA is 1 × 1.
20)
Solve the system by using the inverse of the coefficient matrix.
21) 10x1 - 4x2 = -6
6x1 - x2 = 2
A) (-4, -1) B) (4, 1) C) (1, 4) D) (-1, -4)
21)
22) 2x1 - 6x2 = -6
3x1 + 2x2 = 13
A) (3, 2) B) (2, 3) C) (-3, -2) D) (-2, -3)
22)
Determine whether the matrix is invertible.
23)
A) No B) Yes
24) 6 7
1 18
A) Yes B) No
24)
