Đề thi, bài tập trắc nghiệm online Đại số tuyến tính – Đề 14

1. Which of the following statements is true regarding the eigenvalues of a real symmetric matrix?

A. They are always complex numbers

B. They are always real numbers

C. They are always integers

D. They are always irrational numbers

2. Which of the following operations is NOT an 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 every element in a row

3. What is the purpose of Gram-Schmidt process?

A. To find eigenvalues of a matrix

B. To solve systems of linear equations

C. To orthogonalize a set of linearly independent vectors

D. To compute the determinant of a matrix

4. What is the geometric interpretation of the determinant of a 2x2 matrix formed by two vectors in R^2?

A. The length of the longer vector

B. The area of the parallelogram spanned by the vectors

C. The slope of the line formed by the vectors

D. The distance between the vectors

5. Given two matrices A and B, both of size n x n. Which of the following statements is always true?

A. (A + B)^2 = A^2 + 2AB + B^2

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

C. det(cA) = c * det(A), where c is a scalar

D. det(AB) = det(BA)

6. What is the condition for two vectors u and v to be orthogonal?

A. Their dot product is positive

B. Their dot product is negative

C. Their dot product is zero

D. Their dot product is one

7. Which of the following is NOT a subspace of R^3?

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

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

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

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

8. For a system of linear equations Ax = b, where A is a square matrix, which condition guarantees a unique solution for x?

A. det(A) = 0

B. det(A) ≠ 0

C. b = 0

D. A is a symmetric matrix

9. For a linear transformation T: V → W, what is the relationship between dim(V), dim(ker(T)), and dim(Im(T))?

A. dim(V) = dim(ker(T)) - dim(Im(T))

B. dim(V) = dim(ker(T)) * dim(Im(T))

C. dim(V) = dim(ker(T)) + dim(Im(T))

D. dim(V) = dim(Im(T)) - dim(ker(T))

10. Which of the following is an example of an inner product space?

A. R^n with the Euclidean dot product

B. The set of all n x n matrices with matrix multiplication

C. The set of all polynomials with function composition

D. The set of all continuous functions with pointwise multiplication

11. What is the determinant of a matrix if two of its rows are identical?

A. Always positive

B. Always negative

C. Always zero

D. Cannot be determined

12. In the context of linear transformations, what is the kernel of a transformation T: V → W?

A. The set of all vectors in W that are images of vectors in V

B. The set of all vectors in V that are mapped to the zero vector in W

C. The set of all invertible vectors in V

D. The set of all vectors in W that are linearly independent

13. Given a vector v and a subspace W, what is the orthogonal projection of v onto W?

A. The vector in W that is closest to v

B. The component of v that is perpendicular to W

C. The reflection of v across W

D. The vector obtained by scaling v by a factor related to W

14. What is the primary use of Gaussian elimination?

A. To find eigenvalues and eigenvectors

B. To compute determinants

C. To solve systems of linear equations

D. To orthogonalize vectors

15. Which of the following is a linear transformation from R^2 to R^2?

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

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

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

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

16. If a matrix A is row-equivalent to matrix B, which of the following properties is NOT necessarily preserved?

A. Rank of the matrix

B. Null space of the matrix

C. Solution set of Ax = 0

D. Column space of the matrix

17. If a square matrix A satisfies A^2 = A, then A is called a(n):

A. Invertible matrix

B. Orthogonal matrix

C. Projection matrix

D. Nilpotent matrix

18. Which of the following is a property of orthogonal matrices?

A. Their determinant is always 0

B. Their inverse is equal to their transpose

C. Their eigenvalues are always real

D. Their columns are linearly dependent

19. What does it mean for a matrix to be invertible?

A. Its determinant is zero

B. There exists a matrix that, when multiplied by it, results in the identity matrix

C. All its eigenvalues are zero

D. It is a square matrix

20. What is the dimension of the column space of a matrix A?

A. The number of rows in A

B. The number of columns in A

C. The rank of A

D. The nullity of A

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

A. Any singular matrix

B. Any non-singular matrix

C. Any symmetric matrix

D. Any skew-symmetric matrix

22. If matrix A represents a linear transformation T with respect to basis B, and matrix A' represents the same transformation T with respect to a different basis B', what is the relationship between A and A'?

A. A = A'

B. A is the transpose of A'

C. A and A' are similar matrices

D. A and A' are orthogonal matrices

23. What is the trace of a square matrix?

A. The determinant of the matrix

B. The sum of the eigenvalues of the matrix

C. The product of the diagonal elements

D. The sum of the absolute values of all elements

24. What is the rank of a matrix A if it has 3 rows, 4 columns, and its null space has dimension 2?

A. 1

B. 2

C. 3

D. 4

25. If A and B are invertible n x n matrices, which of the following is the inverse of the product AB?

A. A^(-1)B^(-1)

B. B^(-1)A^(-1)

C. AB^(-1)

D. BA^(-1)

26. What is the characteristic polynomial of a square matrix A used for?

A. To find the determinant of A

B. To find the eigenvalues of A

C. To find the trace of A

D. To find the inverse of A

27. For a symmetric matrix A, what can be said about its eigenvectors corresponding to distinct eigenvalues?

A. They are linearly dependent

B. They are orthogonal

C. They are parallel

D. They are scalar multiples of each other

28. For a square matrix A, if λ is an eigenvalue, what is true about det(A - λI), where I is the identity matrix?

A. det(A - λI) > 0

B. det(A - λI) < 0

C. det(A - λI) = 0

D. det(A - λI) = 1

29. Which of the following is NOT a necessary condition for a set to be considered a vector space over a field F?

A. Closure under vector addition

B. Closure under scalar multiplication

C. Existence of a multiplicative inverse for every non-zero element

D. Existence of an additive identity element

30. Which of the following sets of vectors in R^3 is linearly independent?

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

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

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

D. {(1, 2, 3), (2, 4, 6), (0, 0, 1)}