Deck 6: Linear Transformations

Question 1

Use the function to find the image of

A)
B)
C)
D)
E)

Answer 1

Question 2

Use the function to find the preimage of

A) (-7, 4, t), where t is any real number
B) (4, -7, t), where t is any real number
C) (4, 7, t), where t is any real number
D) (-7, 4, 7)
E) (4, -7, 7)

Answer 2

(4, -7, t), where t is any real number

Question 3

Use the function to find the image of

A) (4, 1)
B) (5, -1)
C) (4, -1)
D) (0, 3)
E) (5, 1)

Answer 3

(4, -1)

Question 4

Use the function to find the preimage of

A) , where t is any real number
B) , where t is any real number
C)
D) , where t is any real number
E)

Question 5

The function is a linear transformation from R³ to R³

Question 6

Let be a linear transformation such that and Find

A)
B)
C)
D)
E)

Question 7

Let be a linear transformation such that and Find

A)
B)
C)
D)
E)

Question 8

The linear transformation is defined by , where Find

A)
B)
C)
D)
E)

Question 9

The linear transformation is defined by , where Find the preimage of .

A) , where r, s, t are real numbers
B) , where r, s, t are real numbers
C) , where r, s, t are real numbers
D) , where r, s, t are real numbers
E) , where r, s, t are real numbers

Question 10

The linear transformation is defined by , where Find

A)
B)
C)
D)
E)

Question 11

Find the kernel of the linear transformation , defined by

A) , where s and t are any real numbers
B) , where r and s are any real numbers
C) , where s is any real number
D) , where r is any real number
E)

Question 12

The linear transformation is defined by , where Find the preimage of (15, 15).

A)
B)
C)
D)
E)

Question 13

Find the kernel of the linear transformation , defined by .

A)
B)
C)
D)
E)

Question 14

Find the kernel of the linear transformation , defined by .

A)
B)
C)
D)
E)

Question 15

The linear transformation is represented by , where Find a basis for the range of T.

A)
B)
C)
D)
E)

Question 16

The linear thransformation is represented by , where Find a basis for the kernel of T.

A)
B)
C)
D)
E)

Question 17

The linear transformation is defined by , where
Find ker(T).

A)
B)
C)
D)
E)

Question 18

The linear transformation is defined by , where Find nullity(T).

A) 3
B) 2
C) 0
D) 1
E) none of these choices

Question 19

The linear transformation is defined by , where Find range(T).

A)
B)
C)
D)
E)

Question 20

The linear transformation is defined by , where Find rank(T)

A) 3
B) 2
C) 4
D) 1
E) 5

Question 21

Let be a linear transformation. Find the nullity of T given that .

A)
B)
C)
D)
E)

Question 22

Let T be a linear transformation from . Given that , find nullity(T).

A)
B)
C)
D)
E)

Question 23

Find the standart matrix A for the linear transformation defined by

A)
B)
C)
D)
E)

Question 24

Use the standard matrix for the linear transformation defined by to find the image of the vector .

A)
B)
C)
D)
E)

Question 25

Find the standard matrix A for the linear transformation T defined by .

A)
B)
C)
D)
E)

Question 26

Find the standard matrix A for the linear transformation T, where T is the counterclockwise rotation of 30° about the origin in .

A)
B)
C)
D)
E)

Question 27

Find the standard matrix A for the linear transformation T, where T is the counterclockwise rotation of 45° about the origin in . Also use A to find for the vector .

A)
B)
C)
D)
E)

Question 28

The standard matrix A for the linear transformation T, where T is the counterclockwise rotation of 45° about the origin in is given by . Use the matrix A to find for the vector .

A)
B)
C)
D)
E)

Question 29

Find the standard matrix A for the linear transformation T where T is the projection onto the vector in .

A)
B)
C)
D)
E)

Question 30

Find the standard matrices for and where , and , .

A)
B)
C)
D)
E)

Question 31

Find where and by using the matrix relative to and .

Question 32

Find where and by using the matrix relative to and .

Question 33

Find the matrix A' for T relative to the basis . ,

A)
B)
C)
D)
E)

Question 34

Find the matrix A' for T relative to the basis . ,

A)
B)
C)
D)
E)

Question 35

Let and be bases for . Find the transition matrix P from B' to B.

A)
B)
C)
D)
E)

Question 36

Let and be bases for , and let be the matrix for relative to B. The transition matrix P from B' to B is . Use the matrices A and P to find and where .

A)
B)
C)
D)
E)

Question 37

Let and be bases for . Find the transition matrix P from B' to B.

A)
B)
C)
D)
E)

Question 38

Let and be bases for , and let be the matrix for relative to B. The transition matrix P from B' to B is . Use the matrices A and P to find and where .

A)
B)
C)
D)
E)

Question 39

Let be the reflection in the x-axis. Find the image of the vector .

A)
B)
C)
D)
E)

Question 40

Let be the reflection in the line . Find the image of the vector .

A)
B)
C)
D)
E)

Question 41

Identify the transformation for an arbitrary vector in the plane.

A) horizontal contraction
B) horizontal expansion
C) vertical contraction
D) vertical expansion
E) horizontal shear

Question 42

The vector v is a fixed point of T if . Find all fixed points of the linear transformation T where T is the reflection in the line y = x.

A) the y-axis
B) the x-axis
C) the line
D) the line y = x
E) none of these choices

Question 43

Sketch the image of the unit square with vertices at (0, 0), (1, 0), (1, 1), and (0, 1) under the transformation T which is the contraction given by .

A)
B)
C)
D)
E)

Question 44

Sketch the image of the unit square with vertices at (0, 0), (1, 0), (1, 1), and (0, 1) under the transformation T which is the contraction given by .

A)
B)
C)
D)
E)

Question 45

Sketch the image of the rectangle with vertices at (0, 0), (0, 4), (1, 4), and (1, 0) under the transformation T which is the contraction given by .

A)
B)
C)
D)
E)

Question 46

Find the image of the Z-shaped figure with vertices (0, 0), (6, 0), (6, 6), and (0, 6) under the transformation T which is the expansion and contraction represented by .

A)
B)
C)
D)
E)