Practice Midterm Test Spring 2021 With Answers
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Subject
Mathematics
Date
Feb 20, 2024
Type
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17
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1.
2.
Student: _____________________ Date: _____________________
Instructor: Judi McDonald Course: Math 220
Assignment: Practice Midterm Test
Spring 2021
ID: 1.1.13
Solve the system.
x
1
−
6x
3
=
15
2x
1
+
4x
2
+
x
3
=
12
2x
2
+
4x
3
=
− 4
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A.
The unique solution of the system is .
,
,
(Type integers or simplified fractions.)
B.
The system has infinitely many solutions.
C.
The system has no solution.
Answer: A. The unique solution of the system is .
3
,
2
,
− 2
(Type integers or simplified fractions.)
ID: 1.1.26
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.
25
20
h
− 5
− 4
1
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The matrix is the augmented matrix of a consistent linear system if h
.
=
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
B.
The matrix is the augmented matrix of a consistent linear system if h
.
≠
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
C.
The matrix is the augmented matrix of a consistent linear system for every value of h.
D.
The matrix is not the augmented matrix of a consistent linear system for any value of h.
Answer: A. The matrix is the augmented matrix of a consistent linear system if h
.
=
− 5
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
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3.
ID: 1.1.28
Determine whether the statement below is true or false. Justify the answer.
Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
Choose the correct answer below.
A.
The statement is false. Interchanging two rows never changes the solution set of the associated linear system. However, scaling a row by a nonzero constant can change the solution set of that system.
B.
The statement is true. Each elementary row operation replaces a system with an equivalent system.
C.
The statement is false. Interchanging two rows never changes the solution set of the associated linear system. However, replacing one row by the sum of itself and a multiple of another row can change the solution set of that system.
D.
The statement is true. Elementary row operations are always applied to an augmented matrix after the solution has been found.
Answer: B. The statement is true. Each elementary row operation replaces a system with an equivalent system.
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4.
ID: 1.2.4
Row reduce the matrix to reduced echelon form. Identify the pivot positions in the final matrix and in the original matrix, and list the pivot columns.
1
2
4
1
2
4
5
− 1
4
5
4
− 2
Row reduce the matrix to reduced echelon form and identify the pivot positions in the final matrix. The pivot positions are indicated by bold values. Choose the correct answer below.
A.
1
0
0
− 1
0
1
0
1
0
0
1
0
B.
1
0
0
1
0
1
0
− 2
0
0
1
1
C.
1
0
0
0
0
1
0
0
0
0
1 1
D.
1
0
0
− 1
0
1
0
1
0
0
1
− 2
Identify the pivot positions in the original matrix. The pivot positions are indicated by bold values. Choose the correct answer below.
A.
1
2
4
1
2
4
5
− 1
4
5
4
− 2
B.
1
2
4
1
2
4
5
− 1
4
5
4
− 2
C.
1
2
4
1
2
4
5
− 1
4
5
4
− 2
D.
1
2
4
1
2
4
5
− 1
4
5
4
− 2
List the pivot columns. Select all that apply.
A.
Column 1
B.
Column 3
C.
Column 4
D.
Column 2
Answers
B. 1
0
0
1
0
1
0
− 2
0
0
1
1
C. 1
2
4
1
2
4
5
− 1
4
5
4
− 2
A. , B. , D. Column 1
Column 3
Column 2
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5.
ID: 1.2.8
Find the general solution of the system whose augmented matrix is given below.
1
4
0
12
2
7
0
19
Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A.
x
1
=
x
2
is free
x
3
is free
B.
x
1
=
x
2
=
x
3
=
C.
x
1
=
x
2
=
x
3
is free
D.
The system has no solution.
Answer: C. x
1
=
− 8
x
2
=
5
x
3
is free
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6.
ID: 1.2.40
What would need to be known about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution
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7.
8.
ID: 1.3.6
Write a system of equations that is equivalent to the given vector equation.
x
1
x
2
x
3
5
− 4
+
6
4
+
− 4
1
=
0
0
Choose the correct answer below.
A.
5x
1
+
6x
2
−
4x
3
= 0
− 4x
1
+
4x
2
= 0
B.
5x
1
+
6x
2
+
4x
3
= 0
4x
1
+
4x
2
+
x
3
= 0
C.
5x
1
+
6x
2
−
4x
3
= 0
− 4x
1
+
4x
2
+
x
3
= 0
D.
5x
1
+
6x
2
+
4x
3
= 0
4x
1
+
4x
2
= 0
Answer:
C. 5x
1
+
6x
2
−
4x
3
= 0
− 4x
1
+
4x
2
+
x
3
= 0
ID: 1.3.15
List five vectors in . Do not make a sketch.
Span
v
1
, v
2
, v
1
=
9
2
− 8
v
2
=
− 7
4
0
List five vectors in . Span
v
1
, v
2
(Use the matrix template in the math palette. Use a comma to separate vectors as needed. Type an integer or a simplified fraction for each vector element. Type each answer only once.)
Answer:
,
,
,
,
0
0
0
− 7
4
0
9
2
− 8
2
6
− 8
− 16
2
8
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9.
10.
ID: 1.4.21
Let , , and . Does span ? Why or why not?
v
1
=
1
0
− 1
0
v
2
=
1
− 1
0
0
v
3
=
0
0
− 1
1
v
1
,
v
2
,
v
3
4
Choose the correct answer below.
A.
No. The set of given vectors spans a plane in . Any of the three vectors can be written as a linear combination of the other two.
4
B.
Yes. When the given vectors are written as the columns of a matrix A, A has a pivot position in every row.
C.
Yes. Any vector in except the zero vector can be written as a linear combination of these three vectors.
4
D.
No. When the given vectors are written as the columns of a matrix A, A has a pivot position in only three rows.
Answer: D. No. When the given vectors are written as the columns of a matrix A, A has a pivot position in only three rows.
ID: 1.4.30
Determine whether the statement below is true or false. Justify the answer.
Any linear combination of vectors can always be written in the form A
x
for a suitable matrix A and vector x
.
Choose the correct answer below.
A.
This statement is true. A
x
can be written as a linear combination of vectors because any two vectors can be combined by addition.
B.
This statement is false. A and x
can only be written as a linear combination of vectors if and only if in A
x
b
,
b
is nonzero.
=
C.
This statement is false. A and x
cannot be written as a linear combination because the matrices do not have the same dimensions.
D.
This statement is true. The matrix A is the matrix of coefficients of the system of vectors. Answer: D. This statement is true. The matrix A is the matrix of coefficients of the system of vectors.
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11.
12.
ID: 1.4.36
Let u = , v
, and w
. It can be shown that u
v
w
0. Use this fact (and no row operations) to find and that satisfy the equation .
4
6
− 7
=
3
4
7
=
5
10
− 70
5
− 5
−
=
x
1
x
2
4
3
6
4
− 7
7
x
1
x
2
=
5
10
− 70
x
1
=
x
2
=
(Simplify your answers.)
Answers 5
− 5
ID: 1.5.33
Determine whether the statement below is true or false. Justify the answer.
The equation x
p
t
v
describes a line through v
parallel to p
.
=
+
Choose the correct answer below.
A.
The statement is false. The effect of adding p
to v
is to move v
in a direction parallel to the plane through p
and 0
. So the equation x
p
t
v
describes a plane through p
parallel to v
.
=
+
B.
The statement is true. The effect of adding p
to v
is to move p
in a direction parallel to the line through v
and 0
. So the equation x
p
t
v
describes a line through v
parallel to p
.
=
+
C.
The statement is false. The effect of adding p
to v
is to move p
in a direction parallel to the plane through v
and 0
. So the equation x
p
t
v
describes a plane through v
parallel to p
.
=
+
D.
The statement is false. The effect of adding p
to v
is to move v
in a direction parallel to the line through p
and 0
. So the equation x
p
t
v
describes a line through p
parallel to v
.
=
+
Answer: D. The statement is false. The effect of adding p
to v
is to move v
in a direction parallel to the line through p
and 0
. So the equation x
p
t
v
describes a line through p
parallel to v
.
=
+
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13.
14.
ID: 1.5.41
A is a 3
3 matrix with three pivot positions.
(a) Does the equation A
x
0
have a nontrivial solution?
=
(b) Does the equation A
x
b
have at least one solution for every possible b
?
=
(a) Does the equation A
x
0
have a nontrivial solution?
=
Yes
No
(b) Does the equation A
x
b
have at least one solution for every possible b
?
=
No
Yes
Answers No
Yes
ID: 1.7.4
Determine if the vectors are linearly independent.
, v
1
=
2
− 1
v
2
=
− 6
3
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The vectors are not linearly independent because if and 1, both not
zero, then .
c
1
=
c
2
=
c
+ c
=
1
v
1
2
v
2
0
B.
The vectors are linearly independent because the vector equation has only the trivial solution.
x
+ x
=
1
v
1
2
v
2
0
Answer: A. The vectors are not linearly independent because if and 1, both not zero, then .
c
1
=
3
c
2
=
c
+ c
=
1
v
1
2
v
2
0
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15.
16.
(1) (2) basic
free
ID: 1.7.11
Find the value(s) of h for which the vectors are linearly dependent. Justify your answer.
, , 3
− 3
6
6
− 9
11
− 2
2
h
The value(s) of h which makes the vectors linearly dependent is(are) because this will cause (1) to be a (2) variable.
(Use a comma to separate answers as needed.)
x
3
x
2
x
1
Answers − 4
(1) x
3
(2) free
ID: 1.7.26
Determine whether the statement below is true or false. Justify the answer.
If x
and y
are linearly independent, and if z
is in Span
, then is linearly dependent.
{
}
x
, y
{
}
x
,
y
, z
Choose the correct answer below.
A.
The statement is true. Since z
is in Span
, z is a linear combination of x
and y
. Since z is a linear combination of x
and y
, the set is linearly dependent.
{
}
x
, y
{
}
x
,
y
, z
B.
The statement is true. Vector z
is in Span
and x
and y
are linearly independent, so z
is a scalar multiple of x
or of y
. Since z is a multiple of x
or y
, the set is linearly dependent.
{
}
x
, y
{
}
x
,
y
, z
C.
The statement is false. Since z
is in Span
, z
cannot be written as a linear combination of x
and y
. The set is linearly independent.
{
}
x
, y
{
}
x
,
y
, z
D.
The statement is false. Vector z
is in Span
and x
and y
are linearly independent, so z
must also be linearly independent of x
and y
. The set is linearly independent.
{
}
x
, y
{
}
x
,
y
, z
Answer: A. The statement is true. Since z
is in Span
, z is a linear combination of x
and y
. Since z is a linear combination of x
and y
, the set is linearly dependent.
{
}
x
, y
{
}
x
,
y
, z
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17.
18.
ID: 1.7.34
Suppose A is a 5
7 matrix. How many pivot columns must A have if its columns span ? Why?
5
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The matrix must have pivot columns. Otherwise, the equation A
would have a free variable, in which case the columns of A would not span .
x
=
0
5
B.
The matrix must have pivot columns. The statements "A has a pivot position in every row" and "the columns of A span " are logically equivalent.
5
C.
The matrix must have pivot columns. If A had fewer pivot columns, then the equation A
would have only the trivial solution.
x
=
0
D.
The columns of a 5
7 matrix cannot span because having more columns than rows makes the columns of the matrix dependent.
5
Answer: B. The matrix must have pivot columns. The statements "A has a pivot position in every row" and "the columns of A span " are logically equivalent.
5
5
ID: 1.8.2
Let A
, u
, and v
. Define T: by T(
x
)
A
x
. Find T(
u
) and T(
v
).
=
1
3
0
0
0
1
3
0
0
0
1
3
=
9
12
− 15
=
b
g
d
3
3
=
T(
u
) =
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
T(
v
) =
(Simplify your answers. Use integers or fractions for any numbers in the expression.)
Answers
3
4
− 5
b
3
g
3
d
3
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19.
20.
ID: 1.8.9
Find all x
in that are mapped into the zero vector by the transformation for the given matrix A.
4
A
x
x
A =
1
− 5
11
− 14
0
1
− 4
4
2
− 8
14
− 20
Select the correct choice below and fill in the answer box(es) to complete your choice.
A.
There is only one vector, which is x
.
=
B.
x
3
x
4
+
C.
x
1
x
2
x
4
+
+
D.
x
3
Answer: B. x
3
x
4
9
4
1
0
+
− 6
− 4
0
1
ID: 1.8.27
Determine whether the statement below is true or false. Justify the answer.
Every linear transformation is a matrix transformation.
Choose the correct answer below.
A.
The statement is false. A matrix transformation is not a linear transformation because multiplication of a matrix A by a vector x
is not linear.
B.
The statement is true. Every linear transformation T(
x
) can be expressed as a multiplication of a matrix A by a vector x
such as A
x
.
C.
The statement is true. Every linear transformation T(
x
) can be expressed as a multiplication of a vector A by a matrix x
such as A
x
.
D.
The statement is false. A matrix transformation is a special linear transformation of the form where A is a matrix.
A
x
x
Answer: D. The statement is false. A matrix transformation is a special linear transformation of the form where A is a matrix.
A
x
x
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21.
(1) (2) (3) (4) is not
is
Let be the transformation that projects each vector x
(x
1
, x
2
, x
3
) onto the plane , so T(
x
)
(x
1
, 0, x
3
). Show that T is a linear transformation.
T : 3
3
=
x
= 0
2
=
The first property for T to be linear is T(
0
)
.
=
Check if this property is satisfied for T.
T(x
1
, x
2
, x
3
) = (x
1
, 0, x
3
)
T(0,0,0) = (
,
,
)
So, is the first property satisfied?
Yes
No
The second property for T to be linear is T(c
u
+d
v
)
(1) for all vectors u
, v in the domain of T and all scalars c, d.
=
Check if this property is satisfied for T. Let u
(u
1
, u
2
, u
3
) and v
(v
1
, v
2
, v
3
). =
=
T(c
+ d )
u
v
= (cu
1
dv
1
, 0, cu
3
dv
3
)
+
+
=
(cu
1
, , )
(dv
1
, , )
+
Factor out the scalar in each ordered triple.
T(c
+ d )
u
v
=
(u
1
, 0, u
3
)
(v
1
, 0, v
3
)
+
Further simplify the previous equation.
T(c
+ d )
u
v
=
c (2) d (3) +
So, is the second property satisfied?
No
Yes
Thus, T (4) linear.
dT( ) + cT( )
u
v
dT( ) − cT( )
u
v
cT( ) − dT( )
u
v
cT( ) + dT( )
u
v
T( )
v
T( )
u
T( )
u
T( )
v
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22.
ID: 1.8.44
Answers 0
0
0
0
Yes
(1) cT(
) + dT( )
u
v
0
cu
3
0
dv
3
c
d
(2) T(
)
u
(3) T(
)
v
Yes
(4) is
ID: 1.9.1
Assume that T is a linear transformation. Find the standard matrix of T.
T: , ( , , , ), and (
, , 0, 0), where (1,0) and (0,1).
2
4
T
e
1
= 9 1 9 1
T
e
2
=
− 6 5
e
1
=
e
2
=
A
(Type an integer or decimal for each matrix element.)
=
Answer:
9
− 6
1
5
9
0
1
0
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23.
ID: 1.9.34
Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer.
T: , T(
)
(
,
), T(
)
( ,
), and T(
)
(
,
), where , , are the columns of the 3
3 identity matrix.
3
2
e
1
= 1 4
e
2
= 2 − 10
e
3
=
− 3 6
e
1
e
2
e
3
a. Is the linear transformation one-to-one?
A.
T is one-to-one because T(
x
)
0
has only the trivial solution.
=
B.
T is not one-to-one because the columns of the standard matrix A are linearly independent.
C.
T is not one-to-one because the standard matrix A has a free variable.
D.
T is one-to-one because the column vectors are not scalar multiples of each other.
b. Is the linear transformation onto?
A.
T is onto because the standard matrix A does not have a pivot position for every row. B.
T is not onto because the standard matrix A contains a row of zeros.
C.
T is not onto because the columns of the standard matrix A span .
2
D.
T is onto because the columns of the standard matrix A span .
2
Answers C. T is not one-to-one because the standard matrix A has a free variable.
D. T is onto because the columns of the standard matrix A span .
2
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24.
Compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let A
, B
, C
, and D
.
=
3
0
− 1
3
− 5
2
=
7
− 4
2
1
− 3
− 3
=
1
2
− 3
1
=
3
4
− 1
4
A, B
A, AC, CD
− 2
− 2
Compute the matrix product A. Select the correct choice below and, if necessary, fill in the answer box within your choice.
− 2
A.
A
− 2
=
(Simplify your answer.)
B.
The expression A is undefined because matrices cannot be multiplied by numbers.
− 2
C.
The expression A is undefined because matrices cannot have negative coefficients.
− 2
D.
The expression A is undefined because A is not a square matrix.
− 2
Compute the martrix sum B
A. Select the correct choice below and, if necessary, fill in the answer box within your choice.
− 2
A.
B
A
− 2
=
(Simplify your answer.)
B.
The expression B
A is undefined because A is not a square matrix.
− 2
C.
The expression B
A is undefined because B and A have different sizes.
− 2
D.
The expression B
A is undefined because B and A have different sizes.
− 2
− 2
Compute the matrix product AC. Select the correct choice below and, if necessary, fill in the answer box within your choice.
A.
AC =
(Simplify your answer.)
B.
The expression AC is undefined because the number of columns in A is not equal to the number of rows in C.
C.
The expression AC is undefined because the number of rows in A is not equal to the number of rows in C.
D.
The expression AC is undefined because the number of rows in A is not equal to the number of columns in C.
Compute the matrix product CD. Select the correct choice below and, if necessary, fill in the answer box within your choice.
A.
CD =
(Simplify your answer.)
B.
The expression CD is undefined because matrices with negative entries cannot be multiplied.
C.
The expression CD is undefined because square matrices cannot be multiplied.
D.
The expression CD is undefined because the corresponding entries in C and D are not equal.
2021/3/6
Practice Midterm Test Spring 2021
https://xlitemprod.pearsoncmg.com/api/v1/print/math
17/17
25.
ID: 2.1.1
Answers A. A
− 2
=
− 6
0
2
− 6
10
− 4
(Simplify your answer.)
A. B
A
− 2
=
1
− 4
4
− 5
7
− 7
(Simplify your answer.)
B. The expression AC is undefined because the number of columns in A is not equal to the number of rows in C.
A. CD =
1
12
− 10
− 8
(Simplify your answer.)
ID: 2.2.1
Find the inverse of the matrix.
5
8
9
3
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The inverse matrix is .
(Type an integer or simplified fraction for each matrix element.)
B.
The matrix is not invertible.
Answer: A. The inverse matrix is .
−
1
19
8
57
3
19
−
5
57
(Type an integer or simplified fraction for each matrix element.)