Master Linear Algebra: Enhance Your Mathematical Skills

Question 1

Identify the orthogonal complement of the subspace spanned by $\begin{bmatrix} 1 \ 2 \ end{bmatrix}$.

Accepted Answer

The set of all vectors perpendicular to $\begin{bmatrix} 1 \ 2 \ end{bmatrix}$

Answer

The set of all vectors parallel to $\begin{bmatrix} 2 \ -1 \ end{bmatrix}$

Answer

The set of all vectors in $\mathbb{R}^2$

Answer

The set of all vectors with zero second component

Question 2

Define a linear transformation.

Accepted Answer

A function that preserves addition and scalar multiplication

Answer

A function that is continuous

Answer

A function that is differentiable

Answer

A function that is injective

Question 3

Find the kernel of the linear transformation $T(x, y) = (x + y, x - y)$.

Accepted Answer

The set of all points on the line y = -x

Answer

The set of all points on the line y = x

Answer

The set of all points on the y-axis

Answer

The set of all points on the x-axis

Question 4

Which of the following is NOT a property of the determinant of a matrix?

Accepted Answer

The determinant of a matrix is equal to the determinant of its transpose.

Answer

The determinant is a scalar value.

Answer

The determinant of the identity matrix is 1.

Answer

The determinant of a triangular matrix is equal to the product of its diagonal entries.

Question 5

Identify the invertible matrix among the following:

Accepted Answer

[1 0; 0 1]

Answer

[1 2; 0 0]

Answer

[0 1; 0 0]

Answer

[0 0; 0 0]

Question 6

Check if the following matrix is diagonalizable: [1 2; 0 1].

Accepted Answer

Yes

Answer

No

Question 7

In the context of eigenvalue calculations, what is the primary significance of a matrix's characteristic polynomial?

Accepted Answer

It provides the matrix's eigenvalues.

Answer

It identifies the eigenvectors corresponding to the eigenvalues.

Answer

It determines the matrix's rank.

Question 8

The determinant of a square matrix A is equal to zero. What does this imply about the matrix A?

Accepted Answer

A is singular.

Answer

A is invertible.

Answer

A has at least one non-zero eigenvalue.

Answer

A is a diagonal matrix.

Question 9

The null space of a matrix A is the set of vectors that satisfy:

Accepted Answer

Ax = 0

Answer

Ax = b for any b

Answer

xAx = 0

Answer

x^T A = 0

Question 10

The eigenvalues of a diagonalizable matrix can be found by:

Accepted Answer

Solving the characteristic equation of the matrix

Answer

Finding the roots of the determinant of the matrix

Answer

Computing the trace of the matrix

Answer

Adding the elements of the matrix

Question 11

The orthogonal complement of a subspace W is:

Accepted Answer

The set of all vectors orthogonal to every vector in W

Answer

The set of all vectors in the span of W

Answer

The set of all vectors in the null space of W

Answer

The set of all vectors not in W

Question 12

The Gram-Schmidt process can be used to:

Accepted Answer

Construct an orthonormal basis for a set of vectors

Answer

Find the eigenvectors of a matrix

Answer

Compute the rank of a matrix

Answer

Invert a matrix

Question 13

The determinant of a matrix is a scalar value that is calculated from the matrix's:

Accepted Answer

Elements

Answer

Eigenvalues

Answer

Eigenvectors

Answer

Rank

Question 14

Which of the following is the characteristic equation of a matrix A?

Accepted Answer

det(A - λI) = 0

Answer

det(A + λI) = 0

Answer

det(I - λA) = 0

Answer

det(λA - I) = 0

Question 15

The null space of a matrix A is the set of all vectors x such that:

Accepted Answer

Ax = 0

Answer

Ax = b

Answer

x = 0

Answer

b = 0

Question 16

Which of the statements is not a valid property of matrix operations?

Accepted Answer

Matrices with varying dimensions can be added or subtracted.

Answer

Matrices can be multiplied by scalars.

Answer

The determinant of a square matrix can be used to find its inverse.

Answer

Matrices with the same number of rows and columns can be multiplied.

Question 17

Considering the matrix:
[1, 2, 3]
[4, 5, 6], what is its dimension?

Accepted Answer

2x3

Answer

1x3

Answer

3x1

Answer

3x2

Question 18

Find the rank of the matrix A = [1 2 3; 4 5 6; 7 8 9].

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 19

Calculate the determinant of the matrix A = [2 1; 3 4].

Accepted Answer

5

Answer

-5

Answer

7

Answer

-7

Question 20

Find the eigenvalues of the matrix B = [4 -2; 2 1].

Accepted Answer

λ1 = 3, λ2 = 5

Answer

λ1 = 2, λ2 = 4

Answer

λ1 = 1, λ2 = 6

Answer

λ1 = -3, λ2 = -5

Question 21

Which of the following statements about matrices is **incorrect**?

Accepted Answer

Matrices can be added regardless of their dimensions.

Answer

Matrices can be multiplied if the number of columns in the first matrix equals the number of rows in the second matrix.

Answer

Matrices can be multiplied by scalars.

Answer

Matrices can be inverted if their determinant is non-zero.

Question 22

Calculate the determinant of the matrix  A =  [[2, 1], [3, 4]]

Accepted Answer

5

Answer

-5

Answer

15

Answer

10

Question 23

Find the matrix representation of the linear transformation f(x) = [2x, 3y] in the basis {e1, e2}, where e1 = [1, 0] and e2 = [0, 1]

Accepted Answer

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

Answer

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

Answer

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

Answer

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