Matrix Representation of Linear Transformations: Interactive Learning Quizzes

Question 1

Given a linear transformation T: R² -> R² represented by the matrix A, what is the matrix representation of the transformation T² (T composed with itself)?

Accepted Answer

A²

Answer

A⁻¹

Answer

AA

Answer

Aᵀ

Question 2

Consider the transformation T: R² -> R³ given by T(x, y) = (x + y, x - y, 2x). What is the matrix representation of T?

Accepted Answer

[1 -1 2]

Answer

[1 1]

Answer

[1 -1]

Answer

[1 1 2]

Question 3

Given a linear transformation T: Rⁿ -> Rᵐ, what can be said about the dimensions of the matrix representation of T?

Accepted Answer

It is an m x n matrix.

Answer

It is an n x m matrix.

Answer

It is an n x n matrix.

Answer

It is an m x m matrix.

Question 4

Find the correct matrix representation of the linear transformation that rotates points 90 degrees clockwise about the origin.

Accepted Answer

[[0, -1], [1, 0]]

Answer

[[0, 1], [1, 0]]

Answer

[[1, 0], [0, -1]]

Answer

[[1, 0], [0, 1]]

Question 5

Which of the following is NOT a characteristic of a matrix representation of a linear transformation?

Accepted Answer

It is always invertible

Answer

It preserves linearity

Answer

It is distinctive for a given transformation

Answer

It relies on the choice of basis

Question 6

If the matrix representation of a linear transformation is invertible, what does it imply about the transformation?

Accepted Answer

Invertibility of the transformation

Answer

Non-invertibility of the transformation

Answer

Existence of a unique fixed point

Answer

Existence of infinitely many fixed points

Question 7

For the linear transformation T(x, y) = (2x - 3y, x + y), determine its null space.

Accepted Answer

Line y = 3x

Answer

x-axis

Answer

y-axis

Answer

Line y = -x

Question 8

Consider the matrix representation A of a linear transformation T. If the determinant of A is 0, what does it imply about T?

Accepted Answer

Non-invertibility of T

Answer

Invertibility of T

Answer

Existence of a unique fixed point

Answer

Trivial null space of T

Question 9

Which of the following matrices represents a reflection over the x-axis?

Accepted Answer

[1 0; 0 -1]

Answer

[1 0; 0 1]

Answer

[-1 0; 0 1]

Answer

[0 1; 1 0]

Question 10

Find the matrix representation of the linear transformation T(x, y) = (2x + y, x - 3y).

Accepted Answer

[2 1; 1 -3]

Answer

[1 2; -3 1]

Answer

[2 1; -3 1]

Answer

[1 2; 1 -3]

Question 11

If the product AB results in the zero matrix, what can be inferred about matrices A and B?

Accepted Answer

Either A or B is the zero matrix

Answer

Both A and B are the zero matrix

Answer

Either A or B is invertible

Answer

Both A and B are invertible

Question 12

Which of the following is NOT a property of matrix representations of linear transformations? Identify the statement that does not accurately describe the characteristics of matrix representations.

Accepted Answer

The determinant is always zero.

Answer

The matrix representation is unique for a given transformation.

Answer

The matrix product represents the composition of transformations.

Answer

The trace of the matrix is the sum of the transformation's eigenvalues.

Question 13

Which of the following transformations is NOT represented by a one-to-one matrix? Choose the transformation that does not have a unique matrix representation.

Accepted Answer

Projection onto a subspace

Answer

Rotation in the plane

Answer

Reflection across a line

Question 14

Consider the linear transformation T: R² → R² defined by T(x, y) = (2x + y, x - 3y). Which of the following matrices represents T?

Accepted Answer

[2 1; 1 -3]

Answer

[-3 1; 1 2]

Answer

[3 2; -1 4]

Answer

[1 2; -3 1]

Question 15

Let T: R^3 -> R^3 be a linear transformation with matrix representation A. If A^2 = 0, what can be concluded about T?

Accepted Answer

T is the zero transformation.

Answer

T is an invertible transformation.

Answer

T is a projection transformation.

Answer

T is a rotation transformation.

Question 16

Consider the linear transformation T(x, y) = (x + y, -x + 2y). What is the matrix representation of T with respect to the standard basis?

Accepted Answer

[1 1; -1 2]

Answer

[1 1; 1 2]

Answer

[-1 1; 1 2]

Answer

[-1 -1; -1 2]

Question 17

What is the matrix representation of the linear transformation T: R^2 -> R^2 defined by T(x, y) = (x + y, x - y)?

Accepted Answer

[1 1; 1 -1]

Answer

[-1 1; 1 -1]

Answer

[1 1; -1 1]

Answer

[1 -1; 1 1]

Question 18

If the matrix [A] represents a linear transformation T, what can be concluded if [A] is invertible?

Accepted Answer

T is invertible

Answer

T is not invertible

Answer

T is partially invertible

Answer

T is singular

Question 19

The kernel of a linear transformation associated with matrix [A] consists of vectors that satisfy:

Accepted Answer

[A]x = 0

Answer

[A]x is undefined

Question 20

Which of the following is a key characteristic of the matrix representation of linear transformations?

Accepted Answer

It is dependent on the choice of basis

Answer

It is unique for a given linear transformation

Answer

It is always a square matrix

Answer

It is always symmetric

Question 21

Consider a linear transformation that scales vectors by a factor of 2 in the x-direction and -3 in the y-direction. Which matrix represents this transformation?

Accepted Answer

[2 0; 0 -3]

Answer

[0 2; 0 -3]

Answer

[0 0; 2 -3]

Answer

[2 -3; 0 0]

Question 22

Which matrix best represents the identity transformation in R^n?

Accepted Answer

The n x n identity matrix

Answer

An upper triangular matrix with all diagonal elements equal to 1

Answer

The zero matrix

Answer

A diagonal matrix with all diagonal elements equal to 1

Question 23

Which of the following is a defining property of the matrix representation of a linear transformation?

Accepted Answer

It preserves linear combinations

Answer

It preserves vector lengths

Answer

It preserves vector angles

Answer

It inverts vectors

Question 24

Let A be the matrix representation of a linear transformation T: R^n -> R^m. If T is invertible, then which of the following is true?

Accepted Answer

A is invertible.

Answer

A is singular.

Answer

A has non-trivial null space.

Answer

A is symmetric.

Question 25

Consider the linear transformation T: C -> C defined by T(z) = z^2. Find its matrix representation with respect to the standard basis of C.

Accepted Answer

[(1, 0, 0), (0, 1, 0), (0, 0, 1)]

Answer

[(1, 0, 0), (0, 1, 0), (0, 0, -1)]

Answer

[(1, 0, 0), (0, -1, 0), (0, 0, 1)]

Answer

[(1, 0, 0), (0, 0, 1), (0, 1, 0)]

Question 26

Let A be the matrix representation of a linear transformation T. If T is surjective, then:

Accepted Answer

The column space of A is equal to the range of T.

Answer

The determinant of A is equal to the rank of T.

Answer

The row space of A is equal to the domain of T.

Answer

The null space of A is equal to the kernel of T.

Question 27

Find the matrix representation of the linear transformation T: R^2 -> R^2 that reflects points about the y-axis.

Accepted Answer

[-1, 0; 0, 1]

Answer

[0, 1; 1, 0]

Answer

[1, 0; 0, -1]

Answer

[0, -1; -1, 0]

Question 28

Let A be the matrix representation of a linear transformation T: R^n -> R^m. What is the dimension of the column space of A?

Accepted Answer

Rank of A

Answer

n + m

Answer

n

Answer

min(n, m)

Question 29

Let T: R^3 -> R^2 be a linear transformation. If the matrix representation of T is A, then what is the dimension of the null space of A?

Accepted Answer

Dimension of the kernel of T

Answer

2

Answer

0

Answer

3

Question 30

What is the matrix representation of the linear transformation T: R^2 -> R^2 that projects a point onto the x-axis?

Accepted Answer

[1, 0; 0, 0]

Answer

[1, 0; 0, 1]

Answer

[0, 0; 1, 0]

Answer

[0, 1; 0, 0]

Question 31

Which of the following linear transformations does the matrix representation [2, 1; -3, 2] correspond to?

Accepted Answer

Rotation by an angle whose cosine is 2/√13 and sine is -3/√13, followed by a scaling by a factor of √13.

Answer

Projection onto the y-axis followed by a translation by (-3, 2).

Answer

Reflection about the x-axis followed by a scaling by a factor of 3.

Question 32

Given the linear transformation  T(x, y) = (2x + 3y, x - y), what is its matrix representation?

Accepted Answer

[[2, 3], [1, -1]]

Answer

[[1, 2], [3, 1]]

Answer

[[2, 3], [1, 1]]

Answer

[[1, 3], [2, -1]]

Question 33

Let's say you have two matrices: A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]. What is the matrix representation of the composition T(S(x)) where T is represented by matrix A and S by matrix B?

Accepted Answer

[[19, 22], [43, 50]]

Answer

[[31, 34], [63, 70]]

Answer

[[25, 28], [55, 60]]

Answer

[[13, 16], [29, 36]]

Question 34

Which of the following matrices represents the identity transformation in R^3?

Accepted Answer

[[1, 0, 0], [0, 1, 0], [0, 0, 1]]

Answer

[[0, 1, 0], [1, 0, 0], [0, 0, 1]]

Answer

[[1, 0, 1], [0, 1, 2], [1, 2, 1]]

Answer

[[0, 0, 1], [0, 1, 0], [1, 0, 0]]

Question 35

Consider the linear transformation T: R^3 -> R^2 defined by T(x, y, z) = (x + y, z). What is the null space of its matrix representation?

Accepted Answer

{(x, y, z) | z = 0}

Answer

{(x, y, z) | x = y = 0}

Answer

{(x, y, z) | y = 0}

Answer

{(x, y, z) | x = 0}

Question 36

Which matrix represents the linear transformation that projects a vector in R^3 onto the xy-plane?

Accepted Answer

[[1, 0, 0], [0, 1, 0], [0, 0, 0]]

Answer

[[0, 1, 0], [1, 0, 0], [0, 0, 0]]

Answer

[[1, 0, 0], [0, 0, 1], [0, 1, 0]]

Answer

[[0, 0, 0], [0, 1, 0], [0, 0, 1]]

Question 37

What is the matrix representation of the linear transformation T(x, y) = (2x + y, x - y)?

Accepted Answer

[[2, 1], [1, -1]]

Answer

[[1, 2], [-1, 1]]

Answer

[[1, 2], [1, 1]]

Answer

[[2, -1], [1, 1]]

Question 38

Let A be the matrix representation of a linear transformation T.  If T(u) = v, how can this be represented using matrix multiplication?

Accepted Answer

Au = v

Answer

vA = u

Answer

Av = u

Answer

uA = v

Question 39

What is the matrix representation of the linear transformation that reflects points about the y-axis?

Accepted Answer

[[ -1, 0 ], [ 0, 1]]

Answer

[[ 1, 0 ], [ 0, -1]]

Answer

[[ 0, 1 ], [ 1, 0]]

Answer

[[ 1, 0 ], [ 0, 1]]

Question 40

Given two linear transformations T and U with matrix representations A and B respectively, what is the matrix representation of the composition T o U?

Accepted Answer

AB

Answer

BA

Answer

A + B

Answer

B - A

Question 41

Is the linear transformation T(x, y) = (x + 2y, 3x - y) invertible?

Accepted Answer

Yes

Answer

No

Answer

Only if x = y

Answer

Cannot be determined

Question 42

Find the kernel of the linear transformation T: R^3 -> R^2 represented by the matrix [[1, 2], [3, 4]]

Accepted Answer

Span{(-2, 1, 0))

Answer

Span{(1, 0, 0))

Answer

Span{(0, 1, 0))

Answer

Span{(0, 0, 1))

Question 43

Determine the dimension of the range of the linear transformation T: R^4 -> R^3 represented by the matrix [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]]

Accepted Answer

3

Answer

1

Answer

4

Answer

2

Question 44

Find the eigenvalues of the linear transformation T: R^2 -> R^2 represented by the matrix [[2, 1], [-1, 2]]

Accepted Answer

3, 1

Answer

1, 2

Answer

3, 2

Answer

2, 1

Question 45

Determine whether the linear transformation T: R^3 -> R^3 represented by the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]] is diagonalizable.

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Question 46

Find a basis for the subspace of R^3 spanned by the vectors (1, 2, 3), (4, 5, 6), and (7, 8, 9).

Accepted Answer

{(1, 2, 3)}

Answer

{(1, 2, 3), (4, 5, 6)}

Answer

{(1, 2, 3), (4, 5, 6), (7, 8, 9)}

Answer

{(4, 5, 6), (7, 8, 9)}