Đề 8 – Bài tập, đề thi trắc nghiệm online Đại số tuyến tính

1. Which of the following statements about the rank of a matrix is always true?

A. rank(A) = rank(A^T) always.

B. rank(A) = rank(A^2) always.

C. rank(A) ≥ rank(A^T) sometimes.

D. rank(A) + rank(A^T) = n for an n x n matrix.

2. Which of these transformations is NOT a linear transformation?

A. T(x) = 2x

B. T(x) = x + 1

C. T(x) = -x

D. T(x) = 0

3. For a square matrix A, if A^2 = A, then A is called:

A. Idempotent (Projector).

B. Involutory.

C. Nilpotent.

D. Orthogonal.

4. What is the span of a set of vectors?

A. The intersection of all subspaces containing the vectors.

B. The smallest subspace containing the vectors.

C. The largest subspace contained within the vectors.

D. The set of all linear combinations of the vectors.

5. If a system of linear equations has more unknowns than equations, it is guaranteed to have:

A. A unique solution.

B. No solution.

C. Infinitely many solutions or no solution.

D. Exactly two solutions.

6. If A and B are n x n matrices, and both are invertible, is (A + B) necessarily invertible?

A. Yes, always.

B. No, not necessarily.

C. Yes, if A and B are symmetric.

D. No, only if det(A) = det(B).

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

A. ad + bc

B. ac - bd

C. ad - bc

D. ab - cd

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

A. Nullity.

B. Rank.

C. Determinant.

D. Trace.

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

A. Cramer`s Rule.

B. Gaussian Elimination.

C. Finding the inverse of the matrix and multiplying by the vector b (A^(-1)b).

D. Graphical method.

10. Which of the following sets of vectors in R^2 is linearly independent?

A. {[1, 2], [2, 4]}

B. {[1, 0], [0, 1], [1, 1]}

C. {[1, 0], [0, 1]}

D. {[0, 0], [1, 2]}

11. What is the kernel of a linear 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 vectors in V that are mapped to non-zero vectors in W.

D. The set of all vectors in W that are not images of vectors in V.

12. If a matrix A is orthogonal, what is the value of A^T * A?

A. The zero matrix.

B. The identity matrix.

C. The matrix A itself.

D. The transpose of A.

13. 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 or column of zeros, then det(A) = 0.

14. The null space of a matrix A consists of all vectors x such that:

A. Ax = b for some non-zero vector b.

B. Ax = 0.

C. A^T x = 0.

D. x^T A = 0.

15. What is the trace of a square matrix?

A. The product of the diagonal elements.

B. The determinant of the matrix.

C. The sum of the diagonal elements.

D. The inverse of the determinant.

16. For a system of linear equations Ax = b to be consistent, what condition must be met?

A. det(A) ≠ 0.

B. rank(A) = rank([A|b]).

C. A must be a square matrix.

D. b must be the zero vector.

17. What is an eigenvector of a square matrix A?

A. A vector v such that Av = 0.

B. A vector v such that Av = v.

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

D. Any non-zero vector.

18. For a symmetric matrix, are its eigenvectors corresponding to distinct eigenvalues always orthogonal?

A. Yes, always.

B. No, never.

C. Only if the eigenvalues are positive.

D. Only if the determinant is non-zero.

19. If two vectors u and v are orthogonal, their dot product u · v is:

A. 1

B. -1

C. 0

D. Equal to their magnitudes.

20. 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 solutions to a non-homogeneous linear differential equation.

D. R^3.

21. What is the rank-nullity theorem state for a linear transformation T: V → W?

A. rank(T) + nullity(T) = dim(W).

B. rank(T) - nullity(T) = dim(V).

C. rank(T) + nullity(T) = dim(V).

D. rank(T) * nullity(T) = dim(V) * dim(W).

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

A. Invertible.

B. Non-invertible (singular).

C. Orthogonal.

D. Diagonalizable.

23. If λ is an eigenvalue of matrix A, then for any scalar k, what is an eigenvalue of matrix A + kI?

A. kλ

B. λ - k

C. λ + k

D. λ/k

24. What is the geometric interpretation of a linear transformation from R^2 to R^2 represented by a matrix with determinant -1?

A. Rotation.

B. Scaling.

C. Reflection.

D. Shear.

25. 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 ≥ 0, y ≥ 0, z ≥ 0.

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

D. The set of all vectors (x, y, z) such that x^2 + y^2 + z^2 = 1.

26. Which of the following is a property of eigenvalues?

A. The eigenvalues of a matrix are always integers.

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

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

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

27. If a linear transformation T: V → W is injective (one-to-one), what can be said about the dimension of V and W?

A. dim(V) ≤ dim(W).

B. dim(V) ≥ dim(W).

C. dim(V) = dim(W).

D. dim(V) > dim(W).

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

A. Eigenvectors.

B. Eigenvalues.

C. Determinant.

D. Trace.

29. What is the condition for a square matrix to be diagonalizable?

A. It must be invertible.

B. It must have n linearly independent eigenvectors, where n is the size of the matrix.

C. It must be symmetric.

D. It must have determinant equal to 1.

30. 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.