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

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

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

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

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

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)

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

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

A A. det(A) = 0 B B. det(A) ≠ 0 C C. b = 0 D D. A is a symmetric matrix

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))

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

A A. Always positive B B. Always negative C C. Always zero D D. Cannot be determined

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

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

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

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)

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

A A. Invertible matrix B B. Orthogonal matrix C C. Projection matrix D D. Nilpotent matrix

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

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

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

A A. Any singular matrix B B. Any non-singular matrix C C. Any symmetric matrix D D. Any skew-symmetric matrix

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

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

A A. 1 B B. 2 C C. 3 D D. 4

A A. A^(-1)B^(-1) B B. B^(-1)A^(-1) C C. AB^(-1) D D. BA^(-1)

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

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

A A. det(A - λI) > 0 B B. det(A - λI) < 0 C C. det(A - λI) = 0 D D. det(A - λI) = 1

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

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)}