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Exam 6: Linear Transformations

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Exam 1: Linear Equations25 Questionsadd course
Exam 2: Matrices48 Questionsadd course
Exam 3: Determinants47 Questionsadd course
Exam 4: Vector Spaces100 Questionsadd course
Exam 5: Inner Product Spaces54 Questionsadd course
Exam 6: Linear Transformations46 Questionsadd course
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The standard matrix A for the linear transformation T, where T is the counterclockwise rotation of 45° about the origin in 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 . is given by 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 . . Use the matrix A to find 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 . for the vector 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 . .

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Question 1
Correct Answer:
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A

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

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D

Use the function Use the function to find the image of to find the image of Use the function to find the image of

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Question 3
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C

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

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Question 4

Let Let be a linear transformation such that and Find be a linear transformation such that Let be a linear transformation such that and Find and Let be a linear transformation such that and Find Find Let be a linear transformation such that and Find

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Question 5

Find the standard matrices for Find the standard matrices for and where , and , . and Find the standard matrices for and where , and , . where Find the standard matrices for and where , and , . , Find the standard matrices for and where , and , . and Find the standard matrices for and where , and , . , Find the standard matrices for and where , and , . .

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Question 6

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

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Question 7

Identify the transformation Identify the transformation for an arbitrary vector in the plane. for an arbitrary vector Identify the transformation for an arbitrary vector in the plane. in the plane.

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Question 8

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

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Question 9

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

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Question 10

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

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Question 11

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

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Question 12

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

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Question 13

Use the function Use the function to find the preimage of to find the preimage of Use the function to find the preimage of

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Question 14

Find the kernel of the linear transformation Find the kernel of the linear transformation , defined by , defined by Find the kernel of the linear transformation , defined by

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Question 15

Find the kernel of the linear transformation Find the kernel of the linear transformation , defined by . , defined by Find the kernel of the linear transformation , defined by . .

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Question 16

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

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Question 17

The function The function is a linear transformation from R<sup>3</sup> to R<sup>3</sup> is a linear transformation from R3 to R3

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Question 18

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

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Question 19

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 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 . .

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