Mathematics

Linear Algebra

449 Questions

Linear algebra involves the study of matrices, vectors, and linear transformations. Common topics include finding the rank of a matrix, calculating eigenvalues and eigenvectors, and performing LU decomposition. These advanced mathematical concepts are frequently tested in engineering, statistics, and civil service examinations.

Matrix rank calculationLU decompositionEigenvalues and eigenvectorsMatrix invertibilityDeterminant properties

Linear Algebra Questions

Multiple choice

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

  1. $$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$
  2. $$\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$$
  3. $$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$
  4. $$\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The matrix $$\begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}$$ represents a reflection about the $x$-axis because it satisfies the following equation: $$\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$$, where $$(x', y')$$ are the coordinates of a point after the reflection and $$(x, y)$$ are the coordinates of the point before the reflection.

Multiple choice

Which of the following matrices represents a projection onto the $x$-axis?

  1. $$\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$$
  2. $$\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$$
  3. $$\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$$
  4. $$\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The matrix $$\begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}$$ represents a projection onto the $x$-axis because it satisfies the following equation: $$\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$$, where $$(x', y')$$ are the coordinates of a point after the projection and $$(x, y)$$ are the coordinates of the point before the projection.

Multiple choice

Which of the following matrices represents a change of basis from the standard basis to the basis $${\begin{bmatrix} 1 \ 2 \end{bmatrix}, \begin{bmatrix} 3 \ 4 \end{bmatrix}}$$?

  1. $$\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$$
  2. $$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$
  3. $$\begin{bmatrix} 2 & 3 \\ 4 & 1 \end{bmatrix}$$
  4. $$\begin{bmatrix} 3 & 2 \\ 4 & 1 \end{bmatrix}$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The matrix $$\begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$$ represents a change of basis from the standard basis to the basis $${\begin{bmatrix} 1 \ 2 \end{bmatrix}, \begin{bmatrix} 3 \ 4 \end{bmatrix}}$$ because it satisfies the following equation: $$\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$$, where $$(x', y')$$ are the coordinates of a point in the new basis and $$(x, y)$$ are the coordinates of the point in the old basis.

Multiple choice

Which of the following matrices represents the transpose of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$?

  1. $$\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$$
  2. $$\begin{bmatrix} 3 & 1 \\ 4 & 2 \end{bmatrix}$$
  3. $$\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}$$
  4. $$\begin{bmatrix} 4 & 2 \\ 3 & 1 \end{bmatrix}$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The matrix $$\begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$$ represents the transpose of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ because it is obtained by interchanging the rows and columns of the original matrix.

Multiple choice

Which of the following matrices represents the determinant of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$?

  1. $$-2$$
  2. $$2$$
  3. $$-10$$
  4. $$10$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The determinant of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$ is $$-2$$ because it is calculated by subtracting the product of the elements on the main diagonal from the product of the elements on the other diagonal.

Multiple choice

Which of the following matrices represents the trace of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$?

  1. $$7$$
  2. $$9$$
  3. $$11$$
  4. $$13$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The trace of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$ is $$7$$ because it is calculated by adding the elements on the main diagonal.

Multiple choice

Which of the following matrices represents the rank of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \ 5 & 6 \end{bmatrix}$$?

  1. $$1$$
  2. $$2$$
  3. $$3$$
  4. $$4$$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The rank of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \ 5 & 6 \end{bmatrix}$$ is $$2$$ because it is the maximum number of linearly independent rows or columns in the matrix.

Multiple choice

Which of the following matrices represents the null space of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$?

  1. $$\{\begin{bmatrix} -2 \\ 1 \end{bmatrix}\}$$
  2. $$\{\begin{bmatrix} 2 \\ -1 \end{bmatrix}\}$$
  3. $$\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}\}$$
  4. $$\{\begin{bmatrix} 0 \\ 0 \end{bmatrix}\}$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The null space of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ is $${\begin{bmatrix} -2 \ 1 \end{bmatrix}}$$ because it is the set of all vectors that are orthogonal to the row space of the matrix.

Multiple choice

Which of the following matrices represents the column space of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$?

  1. $$\{\begin{bmatrix} 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 4 \end{bmatrix}\}$$
  2. $$\{\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 4 \end{bmatrix}\}$$
  3. $$\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}\}$$
  4. $$\{\begin{bmatrix} 0 \\ 0 \end{bmatrix}\}$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The column space of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ is $${\begin{bmatrix} 1 \ 3 \end{bmatrix}, \begin{bmatrix} 2 \ 4 \end{bmatrix}}$$ because it is the set of all linear combinations of the columns of the matrix.

Multiple choice

Which of the following matrices represents the row space of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$?

  1. $$\{\begin{bmatrix} 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 4 \end{bmatrix}\}$$
  2. $$\{\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 4 \end{bmatrix}\}$$
  3. $$\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}\}$$
  4. $$\{\begin{bmatrix} 0 \\ 0 \end{bmatrix}\}$$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The row space of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ is $${\begin{bmatrix} 1 \ 2 \end{bmatrix}, \begin{bmatrix} 3 \ 4 \end{bmatrix}}$$ because it is the set of all linear combinations of the rows of the matrix.

Multiple choice

Which of the following matrices represents the eigenvalues of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$?

  1. $$\{2, 5\}$$
  2. $$\{2, 7\}$$
  3. $$\{3, 4\}$$
  4. $$\{3, 5\}$$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The eigenvalues of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$ are $$2$$ and $$7$$ because they are the roots of the characteristic polynomial of the matrix.

Multiple choice

Which of the following matrices represents the eigenvectors of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$?

  1. $$\{\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 4 \end{bmatrix}\}$$
  2. $$\{\begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \end{bmatrix}\}$$
  3. $$\{\begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \end{bmatrix}\}$$
  4. $$\{\begin{bmatrix} 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 4 \\ 5 \end{bmatrix}\}$$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The eigenvectors of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$ are $$\begin{bmatrix} 1 \ 1 \end{bmatrix}$$ and $$\begin{bmatrix} 1 \ 2 \end{bmatrix}$$ because they are the vectors that are not changed by the matrix except for a scalar multiple.

Multiple choice

In a Hilbert space, the inner product of two vectors is:

  1. A complex number

  2. A real number

  3. A vector

  4. A matrix
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a Hilbert space, the inner product of two vectors is a complex number.

Multiple choice

What is the matrix representation of the Hadamard Gate?

  1. $$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix}$$
  2. $$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -i \ i & 1 \end{pmatrix}$$
  3. $$\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$$
  4. $$\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The matrix representation of the Hadamard Gate is $$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix}$$. This matrix represents the transformation that the Hadamard Gate applies to a qubit.

Multiple choice

What is the determinant of a tensor?

  1. The product of the eigenvalues of the tensor

  2. The contraction of the tensor with the metric tensor

  3. The trace of the tensor

  4. The rank of the tensor
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The determinant of a tensor is the product of the eigenvalues of the tensor. It is used to measure the volume of a parallelepiped spanned by the eigenvectors of the tensor.