Đề 12 – Bài tập, đề thi trắc nghiệm online Đại số tuyến tính

1. If a matrix A is diagonalizable, it means that A is similar to:

A. An identity matrix

B. A zero matrix

C. A diagonal matrix

D. An orthogonal matrix

2. Which of the following is a subspace of R^3?

A. The set of all vectors (x, y, z) such that x + y + z = 1.

B. The set of all vectors (x, y, z) such that x = y^2.

C. The set of all vectors (x, y, z) such that x - y = 0 and z = 0.

D. The set of all vectors (x, y, z) such that xyz = 0.

3. What is the dimension of the vector space of all 3x3 symmetric matrices?

A. 3

B. 6

C. 9

D. 12

4. What is the rank-nullity theorem?

A. It relates the determinant of a matrix to its eigenvalues.

B. It relates the rank of a matrix to the dimension of its null space.

C. It relates the eigenvalues of a matrix to its eigenvectors.

D. It relates the invertibility of a matrix to its determinant.

5. For a square matrix A, if det(A) = 0, then A is:

A. Invertible

B. Non-invertible (singular)

C. Orthogonal

D. Symmetric

6. Which of the following is NOT a property of determinants?

A. det(AB) = det(A)det(B)

B. det(A + B) = det(A) + det(B)

C. det(A^T) = det(A)

D. If A has a row of zeros, then det(A) = 0

7. Which of the following sets of vectors forms a basis for R^2?

A. {(1, 0), (2, 0)}

B. {(1, 1), (2, 2)}

C. {(1, 0), (0, 1), (1, 1)}

D. {(1, 0), (0, 1)}

8. What is the determinant of a 2x2 matrix [[a, b], [c, d]]?

A. ad + bc

B. ac - bd

C. ad - bc

D. ab - cd

9. If λ is an eigenvalue of matrix A, then which of the following is an eigenvalue of matrix A^2?

A. 2λ

B. λ/2

C. λ^2

D. √λ

10. Which operation is NOT elementary row operation?

A. Swapping two rows.

B. Multiplying a row by a non-zero scalar.

C. Adding a multiple of one row to another row.

D. Adding a scalar to all elements in a row.

11. If a system of linear equations has more equations than variables, it is called:

A. Underdetermined system

B. Overdetermined system

C. Consistent system

D. Inconsistent system

12. The Gram-Schmidt process is used to:

A. Solve systems of linear equations.

B. Find eigenvalues and eigenvectors.

C. Orthogonalize a set of vectors.

D. Calculate the determinant of a matrix.

13. What is the trace of a square matrix?

A. The determinant of the matrix.

B. The sum of all elements in the matrix.

C. The sum of the diagonal elements of the matrix.

D. The product of the diagonal elements of the matrix.

14. If A is a square matrix and A^2 = A, then A is called:

A. Involutory matrix

B. Idempotent matrix

C. Nilpotent matrix

D. Orthogonal matrix

15. Which of the following statements is always true for matrix multiplication?

A. Matrix multiplication is commutative.

B. Matrix multiplication is associative.

C. Matrix multiplication always results in a square matrix.

D. Matrix multiplication is defined for any two matrices.

16. Which of the following is a linear transformation?

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

B. T(x, y) = (x + 1, y)

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

D. T(x, y) = (xy, x)

17. What is the characteristic polynomial of a matrix A used to find?

A. The determinant of A

B. The trace of A

C. The eigenvalues of A

D. The eigenvectors of A

18. What is an eigenvector of a matrix A?

A. A vector that is transformed into the zero vector when multiplied by A.

B. A vector that, when multiplied by A, only changes in scale, not direction.

C. A vector that is orthogonal to all columns of A.

D. A vector that is in the null space of A.

19. If the determinant of matrix A is 5, what is the determinant of 2A, where A is a 3x3 matrix?

A. 5

B. 10

C. 20

D. 40

20. If A is an orthogonal matrix, then which of the following is true?

A. A = A^T

B. A = A^-1

C. A^-1 = A^T

D. A + A^T = I

21. The dot product of two orthogonal vectors is always:

A. 1

B. -1

C. 0

D. Greater than 0

22. The null space of a matrix A is:

A. The set of all vectors b for which Ax = b has a solution.

B. The set of all vectors x such that Ax = 0.

C. The span of the columns of A.

D. The set of all eigenvalues of A.

23. If vectors v1, v2, ..., vn are linearly dependent, then:

A. The zero vector can only be expressed as a trivial linear combination of v1, v2, ..., vn.

B. At least one of the vectors can be expressed as a linear combination of the others.

C. All vectors must be orthogonal to each other.

D. The determinant of the matrix formed by these vectors as columns is non-zero.

24. What is the geometric interpretation of a system of two linear equations with two variables having no solution?

A. The two lines are identical.

B. The two lines intersect at a single point.

C. The two lines are parallel and distinct.

D. The two lines are perpendicular.

25. What does the Invertible Matrix Theorem NOT state?

A. A square matrix is invertible if and only if its determinant is non-zero.

B. A square matrix is invertible if and only if its columns are linearly independent.

C. A square matrix is invertible if and only if its rows span R^n.

D. A square matrix is invertible if and only if it is diagonalizable.

26. What is the rank of a matrix?

A. The number of rows in the matrix.

B. The number of columns in the matrix.

C. The dimension of the column space (or row space) of the matrix.

D. The determinant of the matrix.

27. The column space of a matrix A is:

A. The set of all vectors x such that Ax = 0.

B. The set of all linear combinations of the columns of A.

C. The set of all rows of A.

D. The set of all vectors orthogonal to the rows of A.

28. Which of the following is NOT a vector space?

A. The set of all polynomials of degree less than or equal to 2.

B. The set of all 2x2 matrices.

C. The set of all vectors in R^3 with non-negative components.

D. The set of all real numbers.

29. Which of the following matrices is guaranteed to be diagonalizable?

A. Any invertible matrix

B. Any symmetric matrix

C. Any singular matrix

D. Any skew-symmetric matrix

30. For what values of k are the vectors (1, 2, 3), (0, 1, 2), and (0, 0, k) linearly independent?

A. For all values of k.

B. For k = 0 only.

C. For k ≠ 0 only.

D. For k = 1 only.