Đề thi, bài tập trắc nghiệm online Đại số tuyến tínhĐề 14 – Bài tập, đề thi trắc nghiệm online Đại số tuyến tính Đăng vào 2 Tháng 5, 2026 bởi admin Đề 14 – Bài tập, đề thi trắc nghiệm online Đại số tuyến tính Đề 14 – Bài tập, đề thi trắc nghiệm online Đại số tuyến tính Số câu30Quiz ID14829 Làm bài Câu 1 1. Which of the following statements is true regarding the eigenvalues of a real symmetric matrix? A A. They are always complex numbers B B. They are always real numbers C C. They are always integers D D. They are always irrational numbers Câu 2 2. Which of the following operations is NOT an elementary row operation? A A. Swapping two rows B B. Multiplying a row by a non-zero scalar C C. Adding a multiple of one row to another row D D. Adding a scalar to every element in a row Câu 3 3. What is the purpose of Gram-Schmidt process? A A. To find eigenvalues of a matrix B B. To solve systems of linear equations C C. To orthogonalize a set of linearly independent vectors D D. To compute the determinant of a matrix Câu 4 4. What is the geometric interpretation of the determinant of a 2x2 matrix formed by two vectors in R^2? A A. The length of the longer vector B B. The area of the parallelogram spanned by the vectors C C. The slope of the line formed by the vectors D D. The distance between the vectors Câu 5 5. Given two matrices A and B, both of size n x n. Which of the following statements is always true? A A. (A + B)^2 = A^2 + 2AB + B^2 B B. det(A + B) = det(A) + det(B) C C. det(cA) = c * det(A), where c is a scalar D D. det(AB) = det(BA) Câu 6 6. What is the condition for two vectors u and v to be orthogonal? A A. Their dot product is positive B B. Their dot product is negative C C. Their dot product is zero D D. Their dot product is one Câu 7 7. Which of the following is NOT a subspace of R^3? A A. The set of all vectors (x, y, z) such that x + y + z = 0 B B. The set of all vectors (x, y, z) such that x = y and z = 0 C C. The set of all vectors (x, y, z) such that z = 1 D D. The set of all vectors (x, y, z) such that x = 0, y = 0 Câu 8 8. For a system of linear equations Ax = b, where A is a square matrix, which condition guarantees a unique solution for x? A A. det(A) = 0 B B. det(A) ≠ 0 C C. b = 0 D D. A is a symmetric matrix Câu 9 9. For a linear transformation T: V → W, what is the relationship between dim(V), dim(ker(T)), and dim(Im(T))? A A. dim(V) = dim(ker(T)) - dim(Im(T)) B B. dim(V) = dim(ker(T)) * dim(Im(T)) C C. dim(V) = dim(ker(T)) + dim(Im(T)) D D. dim(V) = dim(Im(T)) - dim(ker(T)) Câu 10 10. Which of the following is an example of an inner product space? A A. R^n with the Euclidean dot product B B. The set of all n x n matrices with matrix multiplication C C. The set of all polynomials with function composition D D. The set of all continuous functions with pointwise multiplication Câu 11 11. What is the determinant of a matrix if two of its rows are identical? A A. Always positive B B. Always negative C C. Always zero D D. Cannot be determined Câu 12 12. In the context of linear transformations, what is the kernel of a transformation T: V → W? A A. The set of all vectors in W that are images of vectors in V B B. The set of all vectors in V that are mapped to the zero vector in W C C. The set of all invertible vectors in V D D. The set of all vectors in W that are linearly independent Câu 13 13. Given a vector v and a subspace W, what is the orthogonal projection of v onto W? A A. The vector in W that is closest to v B B. The component of v that is perpendicular to W C C. The reflection of v across W D D. The vector obtained by scaling v by a factor related to W Câu 14 14. What is the primary use of Gaussian elimination? A A. To find eigenvalues and eigenvectors B B. To compute determinants C C. To solve systems of linear equations D D. To orthogonalize vectors Câu 15 15. Which of the following is a linear transformation from R^2 to R^2? A A. T(x, y) = (x^2, y) B B. T(x, y) = (x + y, x - y) C C. T(x, y) = (x + 1, y) D D. T(x, y) = (xy, x) Câu 16 16. If a matrix A is row-equivalent to matrix B, which of the following properties is NOT necessarily preserved? A A. Rank of the matrix B B. Null space of the matrix C C. Solution set of Ax = 0 D D. Column space of the matrix Câu 17 17. If a square matrix A satisfies A^2 = A, then A is called a(n): A A. Invertible matrix B B. Orthogonal matrix C C. Projection matrix D D. Nilpotent matrix Câu 18 18. Which of the following is a property of orthogonal matrices? A A. Their determinant is always 0 B B. Their inverse is equal to their transpose C C. Their eigenvalues are always real D D. Their columns are linearly dependent Câu 19 19. What does it mean for a matrix to be invertible? A A. Its determinant is zero B B. There exists a matrix that, when multiplied by it, results in the identity matrix C C. All its eigenvalues are zero D D. It is a square matrix Câu 20 20. What is the dimension of the column space of a matrix A? A A. The number of rows in A B B. The number of columns in A C C. The rank of A D D. The nullity of A Câu 21 21. Which of the following matrices is guaranteed to be diagonalizable? A A. Any singular matrix B B. Any non-singular matrix C C. Any symmetric matrix D D. Any skew-symmetric matrix Câu 22 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 = A' B B. A is the transpose of A' C C. A and A' are similar matrices D D. A and A' are orthogonal matrices Câu 23 23. What is the trace of a square matrix? A A. The determinant of the matrix B B. The sum of the eigenvalues of the matrix C C. The product of the diagonal elements D D. The sum of the absolute values of all elements Câu 24 24. What is the rank of a matrix A if it has 3 rows, 4 columns, and its null space has dimension 2? A A. 1 B B. 2 C C. 3 D D. 4 Câu 25 25. If A and B are invertible n x n matrices, which of the following is the inverse of the product AB? A A. A^(-1)B^(-1) B B. B^(-1)A^(-1) C C. AB^(-1) D D. BA^(-1) Câu 26 26. What is the characteristic polynomial of a square matrix A used for? A A. To find the determinant of A B B. To find the eigenvalues of A C C. To find the trace of A D D. To find the inverse of A Câu 27 27. For a symmetric matrix A, what can be said about its eigenvectors corresponding to distinct eigenvalues? A A. They are linearly dependent B B. They are orthogonal C C. They are parallel D D. They are scalar multiples of each other Câu 28 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 A. det(A - λI) > 0 B B. det(A - λI) < 0 C C. det(A - λI) = 0 D D. det(A - λI) = 1 Câu 29 29. Which of the following is NOT a necessary condition for a set to be considered a vector space over a field F? A A. Closure under vector addition B B. Closure under scalar multiplication C C. Existence of a multiplicative inverse for every non-zero element D D. Existence of an additive identity element Câu 30 30. Which of the following sets of vectors in R^3 is linearly independent? A A. {(1, 0, 0), (2, 0, 0), (0, 1, 0)} B B. {(1, 1, 0), (0, 1, 1), (1, 2, 1)} C C. {(1, 0, 0), (0, 1, 0), (0, 0, 1)} D D. {(1, 2, 3), (2, 4, 6), (0, 0, 1)} Đề 13 – Bài tập, đề thi trắc nghiệm online Sinh học di truyền Đề 15 – Bài tập, đề thi trắc nghiệm online Công nghệ chế tạo máy
