Linear Transformations: Definition and Properties - Interactive Quiz

Question 1

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

Accepted Answer

Maps linearly independent vectors to linearly dependent vectors

Answer

Preserves vector addition

Answer

Preserves scalar multiplication

Answer

Maps the zero vector to the zero vector

Question 2

Let V and W be vector spaces, and let T: V -> W be a linear transformation. If V has dimension m and W has dimension n, what is the maximum possible rank of T?

Accepted Answer

min(m, n)

Answer

m

Answer

n

Answer

m + n

Question 3

Which of the following is an example of a non-linear transformation?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 4

Let T: R^2 -> R^3 be a linear transformation. If the image of T is a subspace of R^3, then

Accepted Answer

T is a projection.

Answer

T is a reflection.

Answer

T cannot be one-to-one.

Answer

T must be onto.

Question 5

Which definition best describes a linear transformation?

Accepted Answer

A function that preserves both vector addition and scalar multiplication

Answer

A function that preserves vector addition but not scalar multiplication

Answer

A function that preserves scalar multiplication but not vector addition

Answer

A function that preserves neither vector addition nor scalar multiplication

Question 6

Define a linear transformation from vector space V to vector space W.

Accepted Answer

A function T: V -> W that preserves vector addition and scalar multiplication

Answer

A function T: V -> W that preserves vector multiplication

Answer

A function T: V -> W that is one-to-one

Answer

A function T: V -> W that is onto

Question 7

Considering the function T(x) = 2x, determine if T is a linear transformation:

Accepted Answer

Yes

Answer

No

Question 8

Given a linear transformation T: V → W and vectors v1, v2 ∈ V, what is T(v1 + v2) equal to?

Accepted Answer

T(v1) + T(v2)

Answer

T(v1) - T(v2)

Question 9

Given a linear transformation T: V → W and a scalar c, how is T(cv) expressed?

Accepted Answer

cT(v)

Answer

T(cv)

Question 10

Consider the linear transformation T: R^3 → R^2 given by T(x, y, z) = (x + y, y - z). Find the image of the vector (1, -1, 2).

Accepted Answer

(1, -3)

Answer

(0, 0)

Question 11

If a linear transformation T has a nontrivial null space, what can be concluded?

Accepted Answer

T is not invertible

Answer

T is invertible

Question 12

In the context of matrix transformations, identify an example of a linear transformation:

Accepted Answer

Adding a constant to each element of a vector

Answer

Reflecting a vector over a plane

Question 13

Which of the following properties is NOT a consequence of the linearity of a transformation?

Accepted Answer

The transformation maps parallel lines to non-parallel lines

Answer

The transformation preserves the origin

Question 14

Which of the following functions satisfies the definition of a linear transformation?

Accepted Answer

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

Answer

f(x) = x^2

Answer

f(x) = |x|

Answer

f(x) = sin(x)

Question 15

Which of the following precisely defines a linear transformation?

Accepted Answer

A transformation that preserves vector addition and scalar multiplication operations.

Answer

A transformation that maps vectors in one vector space to vectors in the same or a different vector space.

Answer

A transformation that is continuous and differentiable.

Answer

A transformation that has an inverse.

Question 16

Which property is NOT characteristic of linear transformations?

Accepted Answer

They are always injective (one-to-one).

Answer

They preserve vector addition.

Answer

They preserve scalar multiplication.

Answer

They map the zero vector to the zero vector.

Question 17

Identify the function that exemplifies a linear transformation:

Accepted Answer

f(x) = Ax, where A is a matrix

Answer

f(x) = x^2

Answer

f(x) = |x|

Answer

f(x) = e^x

Question 18

Determine the matrix representation of the linear transformation f(x) = 2x + 3 under the standard basis:

Accepted Answer

[2 3]

Answer

[3 2]

Answer

[2 0]

Answer

[0 3]

Question 19

Define the kernel of the linear transformation f(x) = Ax, where A is a 2x2 matrix:

Accepted Answer

The collection of all vectors x such that Ax = 0

Answer

The collection of all vectors x such that f(x) = 0

Answer

The collection of all vectors x such that f(x) = 1

Answer

The collection of all vectors x such that Ax = 1

Question 20

Identify the range of the linear transformation f(x) = (x, 2x, 3x):

Accepted Answer

The collection of all vectors (a, b, c) such that a = 2b and b = 3c

Answer

The collection of all vectors (a, b, c) such that a = b and b = c

Answer

The collection of all vectors (a, b, c) such that a + b + c = 0

Answer

The collection of all vectors (a, b, c) such that a - b + c = 0

Question 21

For the linear transformation f(x) = Ax, where A is an invertible matrix, determine its inverse:

Accepted Answer

f^-1(x) = A^-1x

Answer

f^-1(x) = xA^-1

Answer

f^-1(x) = -Ax

Answer

f^-1(x) does not exist

Question 22

Identify a common misconception about linear transformations:

Accepted Answer

Linear transformations preserve the magnitude of vectors.

Answer

Linear transformations are exclusively defined on finite-dimensional vector spaces.

Answer

Linear transformations always possess an inverse.

Answer

Linear transformations are invariably continuous.

Question 23

Define a linear transformation.

Accepted Answer

A function that maintains vector space operations

Answer

A function that preserves vector norms

Answer

A function that maps vectors to scalars

Answer

A function that maps lines to planes

Question 24

For a linear transformation to be well-defined, is it essential that both its domain and codomain are vector spaces?

Accepted Answer

Yes, both the domain and codomain must be vector spaces.

Answer

No, only the domain needs to be a vector space.

Answer

No, only the codomain needs to be a vector space.

Answer

No, neither the domain nor codomain needs to be a vector space.

Question 25

How does the linearity of a transformation impact the preservation of vector space operations?

Accepted Answer

Linearity preserves both vector addition and scalar multiplication operations.

Answer

Linearity only preserves scalar multiplication, not vector addition.

Answer

Linearity is unrelated to vector space operations.

Answer

Linearity preserves the perpendicularity of vectors.

Question 26

In linear transformation T: V -> W, what is the definition of the null space of T?

Accepted Answer

The set of vectors in V that map to the zero vector in W.

Answer

The set of vectors in W that map to the zero vector in V.

Answer

The set of vectors in V that map to a non-zero vector in W.

Answer

The set of vectors in W that map to a non-zero vector in V.

Question 27

Consider the following function: T(x, y) = (-y, x). Determine if T is a linear transformation.

Accepted Answer

Yes, it preserves vector addition and scalar multiplication.

Answer

No, it does not preserve vector addition.

Answer

No, it does not preserve scalar multiplication.

Answer

No, it does not map the zero vector to the zero vector.

Question 28

Consider the linear transformations T: V → W and U: W → X. Which composition is linear?

Accepted Answer

U∘T

Answer

T∘U

Answer

T

Answer

U

Question 29

Given a linear transformation T from vector space V to W over field F, what is the result of applying T to the zero vector 0v?

Accepted Answer

0

Answer

v

Answer

T(v)

Answer

Not defined

Question 30

Which characteristic is NOT a property of a linear transformation?

Accepted Answer

Multiplies the magnitude of vectors

Answer

Additivity

Answer

Homogeneity

Answer

Preserves the zero vector

Question 31

Which condition uniquely characterizes a one-to-one linear transformation T: V -> W?

Accepted Answer

If T(v1) = T(v2), then v1 = v2 for all v1, v2 in V.

Answer

There exist v1, v2 in V such that T(v1) = T(v2) and v1 != v2.

Answer

T(v) != 0 for all v in V.

Answer

T is onto.

Question 32

Which property is NOT characteristic of a linear transformation?

Accepted Answer

Preserves the magnitude of vectors

Answer

Preserves vector addition

Answer

Preserves scalar multiplication

Answer

Maps zero vectors to zero vectors

Question 33

Which property is NOT applicable to a linear transformation?

Accepted Answer

Preserves the norm of vectors

Answer

Preserves vector addition

Answer

Preserves scalar multiplication

Answer

Preserves the zero vector

Question 34

Given the linear transformation T: R³ → R² defined by T(x, y, z) = (x - y, 2y + z), determine its rank.

Accepted Answer

2

Answer

3

Answer

1

Answer

0

Question 35

Which of the following is the precise definition of a linear transformation?

Accepted Answer

A function from one vector space to another that preserves vector addition and scalar multiplication operations

Answer

A function from one vector space to another that preserves only vector addition

Answer

A function from one vector space to another that preserves only scalar multiplication

Question 36

Consider the function f(x, y) = (2x + 3y, -x + y). Determine if f is a linear transformation and provide a justification.

Accepted Answer

Yes, f is a linear transformation as it satisfies both vector addition and scalar multiplication properties.

Answer

No, f is not homogeneous.

Answer

No, f is not additive.

Question 37

If a linear transformation T: V → W is given by the matrix A, calculate the image of a vector v in V under T.

Accepted Answer

Av

Answer

A⁻¹v

Answer

v⁺A

Answer

A⁺v

Question 38

Which of the following is NOT a direct consequence of the linearity of a transformation?

Accepted Answer

The inverse transformation exists and is also linear

Answer

The transformation preserves linear combinations of vectors

Answer

The transformation maps the zero vector to the zero vector

Question 39

Which of the following is the most general form of a linear transformation from a vector space V to a vector space W?

Accepted Answer

T: V → W where T(v + w) = T(v) + T(w) and T(cv) = cT(v) for all v, w ∈ V and c ∈ R

Answer

T: V → W where T(v + w) = T(v) and T(cv) = cT(v) for all v, w ∈ V and c ∈ R

Answer

T: V → W where T(v + w) = T(v) + T(w) and T(cv) = T(v) for all v, w ∈ V and c ∈ R

Question 40

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

Accepted Answer

Preserves inner products

Answer

Maps zero vectors to zero vectors

Answer

Preserves scalar multiples

Answer

Maps linear combinations to linear combinations

Question 41

Consider the transformation T: R² → R² that sends (x, y) to (x + 2, y - 3). Is T a linear transformation?

Accepted Answer

Yes

Answer

No, because it is not injective

Answer

No, because it does not preserve scalar multiplication

Answer

No, because it does not map zero vectors to zero vectors

Question 42

Which of the following is an example of a linear transformation from the vector space of polynomials of degree 2 or less to R?

Accepted Answer

Evaluation at x = 1

Answer

Integration

Answer

Differentiation

Answer

Finding the zeros

Question 43

Which of the following is the range of the linear transformation T: R³ → R² that sends (x, y, z) to (x + y, x - z)?

Accepted Answer

{(x, y) ∈ R² | x + y = x - z for some z ∈ R}

Answer

{(x, y) ∈ R² | x + y - z = 0 for some z ∈ R}

Answer

{(x, y) ∈ R² | x - y = 0}

Answer

{(x, y) ∈ R² | x + y = 0}

Question 44

Consider the linear transformation T: V → W. If the dimension of V is m and the dimension of W is n, what is the maximum possible rank of T?

Accepted Answer

min(m, n)

Answer

max(m, n)

Answer

m + n

Answer

m - n

Question 45

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

Accepted Answer

T(u + v) = T(u) + T(v) + w, for some non-zero vector w in W

Answer

T(u + v) = T(u) + T(v) for all vectors u, v in V

Answer

T(cu) = cT(u) for all scalars c

Question 46

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

Accepted Answer

Yes, T satisfies both properties of a linear transformation.

Answer

No, T does not satisfy T(u + v) = T(u) + T(v) for all vectors u, v.

Answer

No, T does not satisfy T(cu) = cT(u) for all scalars c.

Question 47

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

Accepted Answer

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

Answer

T(x, y, z) = (x², y + z)

Answer

T(x, y, z) = (x, y, z + 1)

Answer

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

Question 48

Let T: V → W be a linear transformation. If {v₁, v₂, v₃} is a linearly independent set in V, what can we say about {T(v₁), T(v₂), T(v₃)} in W?

Accepted Answer

It may be linearly independent or linearly dependent.

Answer

It is always linearly dependent.

Answer

It is always linearly independent.

Question 49

Given that T: R² → R² is a linear transformation such that T(1, 0) = (2, 1) and T(0, 1) = (-1, 3), find T(3, 2).

Accepted Answer

(5, 9)

Answer

(3, 4)

Answer

(1, 5)

Answer

(6, 1)

Question 50

Which of the following is **NOT** a necessary condition for a linear transformation T: V → W to be invertible?

Accepted Answer

T must be onto.

Answer

T must preserve linear combinations.

Answer

T must be one-to-one.

Answer

The null space of T must be {0}.

Question 51

Let T: R³ → R³ be the linear transformation defined by T(x, y, z) = (x + y, y - z, 2z). What is the dimension of the null space of T?

Accepted Answer

1

Answer

3

Answer

0

Answer

2

Question 52

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

Accepted Answer

The composition of two linear transformations is always a linear transformation.

Answer

The composition of two linear transformations is never a linear transformation.

Answer

The composition of two linear transformations is only a linear transformation if both transformations are invertible.

Question 53

The transformation T: R² → R² defined by T(x, y) = (x, 0) is a projection onto the ____.

Accepted Answer

x-axis

Answer

y-axis

Answer

origin

Question 54

Consider the linear transformation T: R² → R² that rotates every vector in R² counterclockwise by 90 degrees. What is the image of the vector (1, 1) under T?

Accepted Answer

(-1, 1)

Answer

(1, -1)

Answer

(1, 0)

Answer

(0, 1)