Đề 6 – Bài tập, đề thi trắc nghiệm online Đại số tuyến tính

1. Which method is generally most efficient for solving large systems of linear equations?

A. Cramer`s rule.

B. Gaussian elimination.

C. Finding the inverse matrix and multiplying.

D. Graphical method.

2. A subspace must satisfy which of the following conditions?

A. It must contain the zero vector.

B. It must be closed under scalar multiplication.

C. It must be closed under vector addition.

D. All of the above.

3. If A and B are matrices, when is the product AB defined?

A. Always, if A and B are square matrices.

B. Only if A and B have the same dimensions.

C. If the number of columns of A is equal to the number of rows of B.

D. If the number of rows of A is equal to the number of columns of B.

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

A. Consistent system.

B. Inconsistent system.

C. Overdetermined system.

D. Underdetermined system.

5. What is the determinant of the identity matrix of size n x n?

A. 0

B. 1

C. n

D. n!

6. What is the geometric interpretation of the determinant of a 2x2 matrix?

A. The length of the longest eigenvector.

B. The area scaling factor of the linear transformation represented by the matrix.

C. The volume scaling factor of the linear transformation.

D. The sum of the eigenvalues.

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

A. Determinant of A.

B. Eigenvectors of A.

C. Eigenvalues of A.

D. Inverse of A.

8. For a system of linear equations Ax = b, if det(A) is not zero, what can be said about the solution?

A. There is no solution.

B. There are infinitely many solutions.

C. There is a unique solution.

D. The solution depends on b.

9. Which of the following is a property of eigenvalues of a matrix?

A. The sum of eigenvalues is equal to the determinant of the matrix.

B. The product of eigenvalues is equal to the trace of the matrix.

C. The sum of eigenvalues is equal to the trace of the matrix.

D. Eigenvalues are always integers.

10. The rank of a matrix is defined as:

A. The number of rows in the matrix.

B. The number of columns in the matrix.

C. The dimension of the null space of the matrix.

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

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

A. Invertible.

B. Non-invertible (Singular).

C. Orthogonal.

D. Symmetric.

12. Which of the following is an example of a linear transformation?

A. f(x) = x^2

B. f(x) = sin(x)

C. f(x) = 2x + 3

D. f(x) = 3x

13. Which of the following is NOT a requirement for a set of vectors to be a basis for a vector space V?

A. The set must span V.

B. The set must be linearly independent.

C. The set must contain the zero vector.

D. The set must be a subset of V.

14. If A is an invertible matrix, what is the determinant of A^(-1)?

A. det(A)

B. -det(A)

C. 1/det(A)

D. 0

15. If a matrix A is diagonalizable, it means:

A. A is invertible.

B. A is similar to a diagonal matrix.

C. A is orthogonal.

D. A is symmetric.

16. Which of the following transformations is NOT linear?

A. Rotation.

B. Reflection.

C. Scaling.

D. Translation.

17. The column space of a matrix A is:

A. The set of all linear combinations of the rows of A.

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

C. The null space of A^T.

D. The set of solutions to Ax = 0.

18. The set of all solutions to a non-homogeneous linear system Ax = b (where b is not the zero vector) is:

A. A vector space.

B. A subspace.

C. A linear combination of vectors.

D. A translation of a vector space (affine subspace).

19. What is the null space of a matrix A?

A. The set of all vectors b such that Ax = b has a solution.

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

C. The set of all eigenvalues of A.

D. The set of all eigenvectors of A.

20. What is the relationship between the dimensions of the null space and column space of a matrix A with n columns?

A. dim(Null(A)) + dim(Col(A)) = n

B. dim(Null(A)) = dim(Col(A))

C. dim(Null(A)) - dim(Col(A)) = n

D. dim(Null(A)) * dim(Col(A)) = n

21. If two matrices A and B are row equivalent, then they have the same:

A. Determinant.

B. Eigenvalues.

C. Column space.

D. Null space.

22. Which of the following is always true for a symmetric matrix A?

A. All eigenvalues of A are complex numbers.

B. All eigenvalues of A are real numbers.

C. A is not invertible.

D. A is orthogonal.

23. What does it mean for vectors to be linearly independent?

A. One vector can be written as a scalar multiple of another.

B. They are all orthogonal to each other.

C. None of the vectors can be written as a linear combination of the others.

D. They all have the same magnitude.

24. Which of the following is NOT a basic operation in Gaussian elimination?

A. Swapping two rows.

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

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

D. Multiplying two rows together.

25. What is the dimension of the vector space R^3?

A. 1

B. 2

C. 3

D. Infinite.

26. If matrix A is orthogonal, then which of the following is true about its transpose A^T?

A. A^T = A

B. A^T = -A

C. A^T = A^(-1)

D. A^T = 2A

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

A. ad + bc

B. ac - bd

C. ad - bc

D. bc - ad

28. 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.

29. 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 determinants.

30. What is an eigenvector of a matrix A?

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

B. A vector that, when multiplied by A, results in the zero vector.

C. A non-zero vector v such that Av = λv for some scalar λ.

D. A vector with magnitude 1.