Linear Transformations: Definition, Examples, and Applications

Question 1

How do the eigenvalues and eigenvectors of a linear transformation influence its geometric properties?

Accepted Answer

Eigenvalues determine the scaling of vectors, while eigenvectors define the directions of this scaling.

Answer

Eigenvectors are the only vectors unaffected by the linear transformation.

Answer

Eigenvalues and eigenvectors are only relevant to invertible linear transformations.

Question 2

Let T: V → V be a linear transformation. If T is idempotent (i.e., T² = T), then:

Accepted Answer

T is a projection.

Answer

T is invertible.

Answer

T is a rotation.

Answer

T is a reflection.

Question 3

Which of the following is an example of a linear transformation that is not one-to-one?

Accepted Answer

Projecting a vector onto a subspace.

Answer

Rotating a vector about the origin.

Answer

Multiplying a vector by a diagonal matrix.

Answer

Adding a vector to itself.

Question 4

Identify the function that represents a linear transformation:

Accepted Answer

f(x, y) = x + y

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 5

Identify a necessary condition for a linear transformation to be diagonalizable.

Accepted Answer

Distinct eigenvalues

Answer

Symmetric matrix

Answer

Invertible transformation

Answer

Normal transformation

Question 6

For a linear transformation T: V -> V on a finite-dimensional vector space V, which statement is accurate?

Accepted Answer

T has at most dim(V) eigenvalues.

Answer

T has at least one eigenvalue.

Answer

T has exactly dim(V) eigenvalues.

Answer

T has infinitely many eigenvalues.

Question 7

Which of the following is a defining property of linear transformations?

Accepted Answer

Preserves vector addition and scalar multiplication

Answer

Can be represented by a matrix

Answer

Maps the zero vector to a non-zero vector

Answer

May have a non-unique inverse

Question 8

Let T: V → W be a linear transformation. Which statement is equivalent to T being injective (one-to-one)?

Accepted Answer

The kernel of T is trivial (contains only the zero vector).

Answer

The range of T spans W.

Answer

T has a left inverse.

Answer

T has an inverse.

Question 9

Let T: V → W be a linear transformation. Which condition is sufficient to conclude that T is surjective (onto)?

Accepted Answer

V is finite-dimensional and the range of T has the same dimension as W.

Answer

The kernel of T is trivial.

Answer

T has a left inverse.

Answer

T has an inverse.

Question 10

Let T: V → W and S: W → U be linear transformations. Which of the following is always a linear transformation?

Accepted Answer

Their composition, S ∘ T

Answer

Their sum, T + S

Answer

Their difference, T - S

Answer

Their multiplication, T × S

Question 11

Which of the following is NOT a defining property of linear transformations?

Accepted Answer

Commutativity

Answer

Preservation of vector addition

Answer

Preservation of scalar multiplication

Answer

Preservation of linear combinations

Question 12

Let T: V → V be a linear transformation on a vector space V. Which of the following is sufficient to conclude that T is invertible?

Accepted Answer

T is both injective and surjective.

Answer

T is injective.

Answer

T is surjective.

Answer

T is an isomorphism.

Question 13

When a linear transformation T from vector space V to vector space W is both injective and surjective, what can you conclude?

Accepted Answer

T has a unique inverse.

Answer

T is not invertible.

Answer

T has multiple inverses.

Question 14

Describe how linear transformations can be utilized to solve systems of linear equations. Discuss the benefits and limitations of this method.

Accepted Answer

Linear transformations can be used to solve systems of linear equations by converting them into matrix form and applying row operations.

Answer

Linear transformations can only be used to represent systems of linear equations, not to solve them.

Answer

Linear transformations can be used to simplify systems of linear equations by reducing the number of variables.

Answer

Linear transformations can be applied to any system of linear equations, regardless of its size or complexity.

Question 15

Let T1 and T2 be linear transformations from a vector space V to itself. Prove that if T1 and T2 are both injective, then their composition T2 ∘ T1 is also injective.

Accepted Answer

True

Answer

False

Answer

Not always true

Question 16

If a linear transformation T has a trivial kernel (only the zero vector), why is T injective?

Accepted Answer

Because if T(x) = T(y), then x - y is in the kernel of T, meaning x = y.

Answer

The statement is false; a linear transformation can be injective even with a non-trivial kernel.

Answer

The statement is true because the kernel of a linear transformation is always the zero vector.

Answer

The statement is false; the injectivity of a linear transformation depends on its range.

Question 17

Consider a linear transformation on a finite-dimensional vector space. Which statement is valid?

Accepted Answer

Every linear transformation can be diagonalized over the complex numbers.

Answer

Every linear transformation is invertible.

Answer

Every linear transformation has a real eigenvalue.

Answer

Every linear transformation has a complex eigenvalue.

Question 18

Let T: V → W be a linear transformation. Which of the following is equivalent to T being surjective (onto)?

Accepted Answer

Im(T) = W

Answer

T has an inverse

Answer

T is injective

Answer

Ker(T) = {0}

Question 19

Let T: V → W be a linear transformation. Which of the following statements is true if T is not injective?

Accepted Answer

Ker(T) ≠ {0}

Answer

T is invertible

Answer

T is surjective

Answer

Im(T) = W

Question 20

Let T: V → W be a linear transformation. Which of the following statements is true if T is not invertible?

Accepted Answer

T is either not injective or not surjective

Answer

T is both injective and surjective

Answer

T has a unique inverse

Answer

Ker(T) = {0}

Question 21

Which of the following is NOT a property of linear transformations?

Accepted Answer

Preserves dot product

Answer

Preserves scalar multiplication

Answer

Preserves vector subtraction

Answer

Preserves vector addition

Question 22

Which of the following linear transformations is invertible?

Accepted Answer

T: R² -> R², T(x, y) = (x - y, 2x + y)

Answer

T: R² -> R², T(x, y) = (y, x)

Answer

T: R³ -> R², T(x, y, z) = (x, y)

Answer

T: R² -> R, T(x, y) = x^2

Question 23

Let T: R³ -> R³ be a linear transformation such that T(1, 0, 0) = (2, 1, 0), T(0, 1, 0) = (1, 2, 0), and T(0, 0, 1) = (0, 0, 3). What is the matrix representation of T?

Accepted Answer

[2 1 0; 1 2 0; 0 0 3]

Answer

[0 0 3; 1 2 0; 2 1 0]

Answer

[1 2 0; 2 1 0; 0 0 3]

Question 24

Which transformation is NOT a linear transformation?

Accepted Answer

T(x, y) = (x + 2, y^2)

Answer

T(x, y) = (x - y, x + y)

Answer

T(x, y) = (2x, 3y)

Answer

T(x, y) = (0, 2x + y)

Question 25

A linear transformation T: V → W is injective (one-to-one) if and only if:

Accepted Answer

The kernel of T is {0}

Answer

The dimension of V is equal to the dimension of W

Answer

The range of T is W

Answer

T is surjective (onto)

Question 26

Let T: R³ → R² be defined by T(x, y, z) = (x + y, y - z). What is the rank of T?

Accepted Answer

2

Answer

1

Answer

3

Answer

0

Question 27

If T: V → W is linear and B, C are bases for V, W respectively, the matrix of T relative to B and C:

Accepted Answer

Is unique

Answer

Depends only on the order of vectors in B

Answer

Is not unique

Answer

Depends only on the order of vectors in C

Question 28

Which statement is TRUE about the inverse of a linear transformation?

Accepted Answer

A linear transformation is invertible if and only if it is bijective

Answer

The inverse of a linear transformation is not necessarily linear

Answer

Every linear transformation has an inverse

Answer

The inverse of a linear transformation is unique only if the transformation is from Rⁿ to Rⁿ

Question 29

The composition of linear transformations T: V → W and S: W → U is:

Accepted Answer

A linear transformation from V to U

Answer

Only defined if V = U

Answer

Not necessarily a linear transformation

Answer

A linear transformation from U to V

Question 30

Let T: R² → R² be linear, T(x, y) = (x - y, 2x + y). Which is an eigenvector of T?

Accepted Answer

(1, -1)

Answer

(0, 1)

Answer

(1, 1)

Answer

(2, 1)

Question 31

The null space of a linear transformation T: V → W is:

Accepted Answer

A subspace of V

Answer

A subspace of W

Answer

The same as the range of T

Answer

Not necessarily a subspace

Question 32

If a linear transformation T: V → V is diagonalizable, then:

Accepted Answer

There exists a basis for V consisting of eigenvectors of T

Answer

T is injective (one-to-one)

Answer

T is surjective (onto)

Answer

T is invertible

Question 33

Which is NOT a property of a linear transformation T: V → W?

Accepted Answer

T(u + v) = T(u) + T(v) + c, for a non-zero constant c in W

Answer

T(u + v) = T(u) + T(v)

Answer

T(0) = 0

Answer

T(cu) = cT(u) for any scalar c

Question 34

Let T: R^2 -> R^2 be defined by T(x, y) = (x + y, 2x). Is T a linear transformation?

Accepted Answer

Yes

Answer

No, it does not preserve scalar multiplication

Answer

No, it does not preserve vector addition

Answer

No, it does not satisfy both properties of linearity

Question 35

Let T: R^3 -> R^2 be defined by T(x, y, z) = (x + y, z). What is the kernel of T?

Accepted Answer

{(t, -t, 0) | t ∈ R}

Answer

{(0, 0, 0)}

Answer

{(t, 0, 0) | t ∈ R}

Question 36

Let T: R^2 -> R^2 be defined by T(x, y) = (2x, y). Is T an injective linear transformation?

Accepted Answer

Yes

Answer

No, it does not preserve vector addition

Answer

No, it is not one-to-one

Answer

No, it is not onto

Question 37

Let T: R^2 -> R^2 be defined by T(x, y) = (x + y, x - y). Is T an invertible linear transformation?

Accepted Answer

Yes

Answer

No, it is not one-to-one

Answer

No, it is not both one-to-one and onto

Answer

No, it is not onto

Question 38

Which of the following is NOT a property of an invertible linear transformation?

Accepted Answer

Its kernel is non-zero

Answer

It has an inverse linear transformation

Answer

It maps linearly independent sets to linearly independent sets

Answer

It is both injective and surjective

Question 39

Let T: R^3 -> R^3 be defined by T(x, y, z) = (x + y, y + z, x + z). Find the matrix representation of T with respect to the standard basis.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 40

Let T: R^2 -> R^3 be defined by T(x, y) = (x, y, x + y). Find the rank of T.

Accepted Answer

2

Answer

1

Answer

4

Answer

3

Question 41

Which of the following statements about the composition of linear transformations is TRUE?

Accepted Answer

The composition of two linear transformations is also a linear transformation

Answer

The composition of two linear transformations is not necessarily a linear transformation

Answer

The composition of two linear transformations is always invertible

Question 42

Which of the following is NOT a property of a linear transformation T: V → W?

Accepted Answer

T(u + v) = T(u) + T(v) + T(u, v)

Answer

T(0v) = 0w

Answer

T(cu) = cT(u) for any scalar c

Question 43

Let T: R² → R² be defined by T(x, y) = (x + y, 2x - y). Is T a linear transformation?

Accepted Answer

Yes

Answer

No, it is not a function

Answer

No, it does not satisfy T(u + v) = T(u) + T(v)

Answer

No, it does not satisfy T(cu) = cT(u)

Question 44

Consider the linear transformation T: R³ → R² defined by T(x, y, z) = (x + y, z). What is the null space of T?

Accepted Answer

{(t, -t, 0) | t ∈ R}

Answer

{(t, t, 0) | t ∈ R}

Answer

{(0, 0, 0)}

Answer

{(t, 0, 0) | t ∈ R}

Question 45

Let T: V → W be a linear transformation. Which of the following statements is TRUE?

Accepted Answer

If T is injective, then the null space of T is {0v}

Answer

If T is injective, then the null space of T is W

Answer

If T is surjective, then the null space of T is {0v}

Question 46

Let T: R² → R² be the linear transformation that rotates vectors counterclockwise by 90 degrees. What is the standard matrix representation of T?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 47

Let T: V → W be a linear transformation. Which of the following statements is FALSE?

Accepted Answer

The range of T is always equal to W

Answer

The composition of two linear transformations is a linear transformation

Answer

The inverse of an invertible linear transformation is a linear transformation

Answer

The sum of two linear transformations is a linear transformation

Question 48

What is the rank of the linear transformation T: R³ → R² defined by T(x, y, z) = (x + y, z)?

Accepted Answer

2

Answer

3

Answer

1

Answer

0

Question 49

Which of the following is an example of a linear transformation from R² to R³?

Accepted Answer

T(x, y) = (x + y, 2x, -y)

Answer

T(x, y) = (x², y)

Answer

T(x, y) = (x + y, 2x, -y²)

Answer

T(x, y) = (x + y, 2x + 1, -y)

Question 50

Which of the following is NOT a property of linear transformations?

Accepted Answer

It changes the magnitude of vectors.

Answer

It preserves vector addition.

Answer

It preserves scalar multiplication.

Question 51

Let T: R^2 -> R^3 be a linear transformation. If T(1,2) = (a,b,c), what is T(2,4)?

Accepted Answer

(2a, 4b, 4c)

Answer

(a, 2b, 2c)

Answer

(a+2, b+4, c+4)

Answer

(2a+4, 4b+8, 4c+8)

Question 52

Which of the following linear transformations is injective?

Accepted Answer

T: R^2 -> R^2, T(x,y) = (x-y, x+y)

Answer

T: R^2 -> R^3, T(x,y) = (x, y, 0)

Answer

T: R^3 -> R^2, T(x,y,z) = (x, y)

Answer

T: R^2 -> R^2, T(x,y) = (x^2, y^2)

Question 53

Is the linear transformation T: R^3 -> R^4, T(x,y,z) = (x, y, z, 0) surjective?

Accepted Answer

No

Answer

Yes

Answer

Cannot be determined

Question 54

Which of the following is the matrix representation of the linear transformation T: R^2 -> R^3, T(x,y) = (x+y, 2x-y, 3y)?

Accepted Answer

[[1, 1], [2, -1], [0, 3]]

Answer

[[1, -1], [2, 1], [0, 3]]

Answer

[[1, 1], [2, 0], [0, 3]]

Answer

[[1, 0], [2, 1], [0, 3]]

Question 55

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

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Answer

Depends on the specific values of x and y

Question 56

If T: R^n -> R^m is a linear transformation, what is the dimension of the kernel of T?

Accepted Answer

n-m

Answer

Cannot be determined without more information

Answer

n+m

Answer

m-n

Question 57

Which of the following is an example of a linear transformation that is not a matrix transformation?

Accepted Answer

The derivative operator

Answer

The scaling operator

Answer

The projection operator

Answer

The rotation operator

Question 58

In a linear transformation, if the determinant of the transformation matrix is zero, what can be concluded?

Accepted Answer

The transformation is not invertible.

Answer

The transformation is surjective.

Answer

The transformation is injective.

Answer

The transformation preserves norms.

Question 59

Which of the following is an application of linear transformations in computer graphics?

Accepted Answer

3D transformations

Answer

Signal processing

Answer

Cryptography

Answer

Image enhancement