Invertible Linear Transformations: Definition, Properties, and Examples

Question 1

In image processing, how can invertible linear transformations be used to enhance an image's contrast?

Accepted Answer

Apply a transformation that increases the difference between pixel values, enhancing contrast.

Answer

Use the inverse transformation to restore the original image.

Answer

Invertible transformations cannot enhance contrast as they preserve vector lengths.

Answer

The nullspace of the transformation identifies pixels that do not affect contrast.

Question 2

Let T: R^2 -> R^2 be a linear transformation represented by matrix A. If the determinant of A is 0, then T is:

Accepted Answer

Not invertible

Answer

Invertible

Answer

One-to-one but not onto

Answer

Onto but not one-to-one

Question 3

True or False: Every invertible linear transformation is a bijection.

Accepted Answer

True

Answer

False

Question 4

True or False: The inverse of an invertible linear transformation is unique.

Accepted Answer

True

Answer

False

Question 5

Let T: R^2 -> R^2 be a linear transformation represented by the matrix A. If the determinant of A is 5, what is the determinant of T^-1?

Accepted Answer

1/5

Answer

5

Answer

-5

Answer

1

Question 6

Describe the geometric interpretation of invertibility for linear transformations, connecting it to their kernel and range.

Accepted Answer

Invertible transformations have a trivial kernel and a non-trivial range.

Answer

Invertible transformations have a non-trivial kernel and a trivial range.

Answer

Invertible transformations have both a trivial kernel and a trivial range.

Answer

Invertible transformations have both a non-trivial kernel and a non-trivial range.

Question 7

Let T: R^2 -> R^3 be a linear transformation. If the dimension of the range of T is 2, can T be invertible?

Accepted Answer

No, because the dimension of the range is less than the dimension of the codomain.

Answer

Yes, because the range of T is a subspace of the codomain.

Answer

Yes, because the dimension of the range is equal to the dimension of the domain.

Answer

No, because the dimension of the range is greater than the dimension of the domain.

Question 8

Let T: V -> V be a linear transformation. If T^2 = I, where I is the identity transformation, what can be concluded about T?

Accepted Answer

T is invertible.

Answer

T is not invertible.

Answer

T is symmetric.

Answer

T is orthogonal.

Question 9

Let T: R^3 -> R^3 be a linear transformation such that T(1, 0, 0) = (1, 2, -1), T(0, 1, 0) = (3, 1, 0), and T(0, 0, 1) = (2, 0, 1). Find the matrix representation of T.

Accepted Answer

A = 
[1 3 2
2 1 0
-1 0 1]

Answer

A = 
[1 3 2
2 1 1
-1 0 2]

Answer

A = 
[1 3 2
2 0 1
-1 1 0]

Answer

A = 
[1 3 2
2 0 0
-1 1 1]

Question 10

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

Accepted Answer

Yes, because T is one-to-one and onto.

Answer

No, because T is not one-to-one.

Answer

No, because T is not onto.

Answer

Cannot determine from the given information.

Question 11

Which of the following transformations is NOT invertible?

Accepted Answer

Projection onto the xz-plane.

Answer

Reflection about the y-axis.

Answer

Rotation by 90 degrees around the z-axis.

Answer

Scaling by a factor of 2.

Question 12

Which of the following matrices represents an invertible linear transformation?

Accepted Answer

[2 1; 0 1]

Answer

[1 0; 0 0]

Answer

[0 1; 0 0]

Answer

[1 1; 1 1]

Question 13

**Question:** If an invertible linear transformation is represented by the matrix A and A is diagonalizable, describe the steps to find A's inverse.

Accepted Answer

**Option 1:** Find the eigenvalues and eigenvectors of A. Construct the diagonal matrix D and the invertible matrix P such that A = PDP^-1. Then, A^-1 = P^-1D^-1P.

Answer

**Option 2:** Reduce A to the identity matrix I using row operations. The matrix of row operations will be A^-1.

Answer

**Option 3:** Substitute the values of A into the formula A^-1 = (1/det(A)) * A^T.

Answer

**Option 4:** An invertible linear transformation does not have an inverse if it is diagonalizable.

Question 14

Which single condition ensures that a linear transformation T: V → W is invertible?

Accepted Answer

T is both one-to-one and onto

Answer

T is one-to-one

Answer

T is onto

Question 15

If A is an n x n matrix, then the corresponding linear transformation T(x) = Ax is invertible if and only if:

Accepted Answer

det(A) ≠ 0

Answer

det(A) = 0

Question 16

Which matrix represents an invertible linear transformation?

Accepted Answer

[1 0; 0 1]

Answer

[1 2; 3 4]

Question 17

Determine whether the transformation T(x) = Ax is invertible, where A = [[2, -1], [-3, 2]].

Accepted Answer

Yes

Answer

No

Answer

Not enough information

Answer

Invertible only if x ≠ 0

Question 18

Which of the following is a necessary and sufficient condition for a linear transformation to be invertible?

Accepted Answer

The determinant of its matrix representation is nonzero.

Answer

The matrix representation is diagonalizable.

Answer

The matrix representation has an odd number of rows.

Question 19

If the rank of a linear transformation is equal to the dimension of its domain, is it invertible?

Accepted Answer

Yes

Answer

No

Answer

Not enough information

Answer

Maybe

Question 20

What is the geometric interpretation of an invertible linear transformation?

Accepted Answer

It preserves lengths and angles.

Answer

It shears points in a particular direction.

Answer

It reflects points across the origin.

Answer

It rotates points around a fixed axis.

Question 21

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

Accepted Answer

Yes

Answer

Not enough information

Answer

Invertible only if x ≠ y

Answer

No

Question 22

What is the inverse of the linear transformation T(x) = [[1, -2], [3, 1]]x?

Accepted Answer

[[1/7, 2/7], [-3/7, 1/7]]

Answer

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

Answer

[[7, -2], [-3, 1]]

Question 23

Which of the following is true about the composition of two invertible linear transformations?

Accepted Answer

It is also invertible.

Answer

It is always invertible.

Answer

It is not necessarily invertible.

Answer

It is invertible only if the transformations commute.

Question 24

Identify the statement that is ALWAYS true about an invertible linear transformation T: V → W, where V and W are vector spaces.

Accepted Answer

T is both one-to-one and onto.

Answer

T maps the zero vector of V to the zero vector of W.

Answer

The range of T is a proper subset of W.

Answer

The kernel of T is equal to W.

Question 25

Consider a square matrix A. If the determinant of A is non-zero, what can be deduced about the linear transformation T(x) = Ax?

Accepted Answer

T is an invertible linear transformation.

Answer

T is a non-invertible linear transformation.

Answer

T is one-to-one but not onto.

Answer

T is onto but not one-to-one.

Question 26

Consider an invertible linear transformation T: R³ → R³ and a subspace S of R³. Which of the following is ALWAYS true?

Accepted Answer

The dimension of T(S) is equal to the dimension of S, i.e., dim(T(S)) = dim(S).

Answer

dim(T(S)) is less than dim(S), i.e., dim(T(S)) < dim(S).

Answer

T(S) is equal to S.

Answer

T⁻¹(S) is a subspace of S.

Question 27

Identify the matrix that corresponds to a non-invertible linear transformation.

Accepted Answer

[[1, 2], [2, 4]]

Answer

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

Answer

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

Answer

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

Question 28

A linear transformation T: R² → R² reflects points across the x-axis. Determine the eigenvalue associated with the eigenvector (1, 0).

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 29

Consider a linear transformation T: P₂ → P₂ defined by T(p(x)) = p'(x), where P₂ is the vector space of polynomials of degree at most 2. Is T invertible? Explain your answer.

Accepted Answer

No, because T is not one-to-one.

Answer

Yes, because T is onto.

Answer

Yes, because the kernel of T is trivial.

Answer

No, because T is not a linear transformation.

Question 30

Given a linear transformation T: R³ → R³ with a standard matrix representation A, if the rank of A is 2, what can be said about the invertibility of T?

Accepted Answer

T is not invertible.

Answer

T is onto but not one-to-one.

Answer

T is invertible.

Answer

T is one-to-one but not onto.

Question 31

In the context of data compression, invertible linear transformations are used for encoding and decoding. Which property of invertible transformations is essential for ensuring the complete recovery of the original data after decoding?

Accepted Answer

The transformation is both one-to-one and onto.

Answer

The transformation preserves the lengths of vectors.

Answer

The transformation preserves the angles between vectors.

Answer

The transformation maps the zero vector to itself.

Question 32

Given a linear transformation T from R² to R² represented by matrix A, which condition guarantees that T is invertible?

Accepted Answer

The determinant of A is non-zero (det(A) ≠ 0).

Answer

A is diagonalizable.

Answer

A is a symmetric matrix.

Answer

A has linearly independent columns.

Question 33

For a linear transformation T from vector space V to W, which statement is equivalent to saying that T is invertible?

Accepted Answer

T is both one-to-one and onto.

Answer

T is onto but not one-to-one.

Answer

T is neither one-to-one nor onto.

Answer

T is one-to-one but not onto.

Question 34

A linear transformation T from R³ to R³ maps the vector (1, 2, 3) to (4, 5, 6) and vice versa. What can we conclude about T?

Accepted Answer

T is invertible.

Answer

T is not invertible.

Answer

T is one-to-one but not onto.

Answer

T is onto but not one-to-one.

Question 35

Which of the following linear transformations is NOT invertible?

Accepted Answer

The projection onto a subspace of Rⁿ.

Answer

The rotation about the origin in R².

Answer

The reflection about a line in R².

Question 36

Which of the following conditions *must* hold for a linear transformation T: V → W to be invertible?

Accepted Answer

T is both one-to-one and onto.

Answer

T is one-to-one.

Answer

T is onto.

Answer

T maps the zero vector in V to the zero vector in W.

Question 37

Let T: R² → R² be the linear transformation defined by T(x, y) = (x + 2y, 3x - y). What is the inverse transformation, T⁻¹?

Accepted Answer

T⁻¹(x, y) = (1/7x + 2/7y, 3/7x - 1/7y)

Answer

T⁻¹(x, y) = (x + 2y, 3x + y)

Answer

T⁻¹(x, y) = (x - 2y, 3x + y)

Question 38

If T: V → W is an invertible linear transformation, what is the relationship between the dimensions of V and W?

Accepted Answer

dim(V) = dim(W)

Answer

dim(V) < dim(W)

Answer

dim(V) > dim(W)

Answer

There is no relationship between the dimensions.

Question 39

Which of the following statements is TRUE regarding the composition of invertible linear transformations?

Accepted Answer

The composition of two invertible linear transformations is also invertible.

Answer

The inverse of a composition is the composition of the inverses, in the same order.

Answer

The composition of two invertible linear transformations is not necessarily invertible.

Question 40

Consider a linear transformation T: R³ → R³. If the matrix representation of T with respect to the standard basis is an invertible matrix, what can we conclude about T?

Accepted Answer

T is invertible.

Answer

T is onto but not one-to-one.

Answer

T is one-to-one but not onto.

Answer

T is not invertible.

Question 41

Suppose T: R³ → R³ is a linear transformation with the property that T(x, y, z) = (x, y, 0). Is T invertible?

Accepted Answer

No, T is not invertible.

Answer

Not enough information to determine.

Answer

Yes, T is invertible.

Answer

It depends on the choice of basis.

Question 42

Let T: V → W be an invertible linear transformation. Which of the following is *always* true?

Accepted Answer

The kernel of T contains only the zero vector.

Answer

The inverse transformation, T⁻¹, is also linear.

Answer

The determinant of the matrix representation of T is zero.

Answer

The range of T is equal to V.

Question 43

Given a system of linear equations Ax = b, where A is an invertible matrix, how many solutions does the system have?

Accepted Answer

Exactly one solution

Answer

No solutions

Answer

The number of solutions depends on the specific values in b.

Answer

Infinitely many solutions

Question 44

Let T: R² → R² be a reflection about the x-axis. Is T invertible? If so, what is T⁻¹?

Accepted Answer

Yes, T⁻¹ is also a reflection about the x-axis.

Answer

No, T is not invertible.

Answer

It depends on the choice of basis.

Answer

Yes, T⁻¹ is a rotation by 180 degrees.

Question 45

If T: V → V is a linear transformation such that T² = I (where I is the identity transformation), is T necessarily invertible?

Accepted Answer

Yes, T is invertible.

Answer

No, T is not necessarily invertible.

Answer

It depends on the dimension of V.

Answer

It depends on the specific choice of T.