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

A crucial property of real symmetric matrices is that all their eigenvalues are real numbers. This is a significant result in linear algebra and has implications in various applications, such as quantum mechanics and vibration analysis. Eigenvalues of real symmetric matrices are guaranteed to be real, though not necessarily integers or irrational.

Elementary row operations are: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another. Adding a scalar to every element in a row is not an elementary row operation, as it does not preserve row equivalence in the same way.

The Gram-Schmidt process is an algorithm used to orthogonalize a set of linearly independent vectors in an inner product space. It takes a set of linearly independent vectors and produces an orthonormal set that spans the same subspace. It's not directly used for eigenvalues, solving linear equations, or determinant computation.

The absolute value of the determinant of a 2x2 matrix formed by two vectors in R^2 is geometrically interpreted as the area of the parallelogram spanned by these two vectors. In R^3, the determinant of a 3x3 matrix formed by three vectors represents the volume of the parallelepiped spanned by them.

Matrix multiplication is not always commutative (AB may not equal BA), so (A+B)^2 expansion is generally not A^2 + 2AB + B^2. Also, det(A+B) is generally not det(A) + det(B), and det(cA) = c^n * det(A) for n x n matrices. However, det(AB) = det(A)det(B) = det(B)det(A) = det(BA), so det(AB) always equals det(BA).

Two vectors u and v are orthogonal (perpendicular) if and only if their dot product (inner product) is zero. A zero dot product signifies that the angle between the vectors is 90 degrees (or one of them is the zero vector).

For a subset to be a subspace, it must contain the zero vector and be closed under addition and scalar multiplication. The set of vectors (x, y, z) such that z = 1 does not contain the zero vector (0, 0, 0), so it's not a subspace. The other sets are subspaces: option 1 and 2 are planes and a line through the origin respectively, and option 4 is just the zero vector, which is also a subspace.

A unique solution exists if and only if the matrix A is invertible. A square matrix A is invertible if and only if its determinant is non-zero (det(A) ≠ 0). If det(A) = 0, there are either no solutions or infinitely many solutions. b=0 and A being symmetric do not guarantee a unique solution.

The Rank-Nullity Theorem states that for a linear transformation T: V → W, the dimension of the domain V is equal to the sum of the dimension of the kernel of T (nullity) and the dimension of the image of T (rank). That is, dim(V) = dim(ker(T)) + dim(Im(T)).

An inner product space is a vector space equipped with an inner product, which is a generalization of the dot product. R^n with the Euclidean dot product is a standard example of an inner product space. Matrix multiplication and function composition are operations, not inner products. Pointwise multiplication of functions is not generally an inner product in the standard sense for defining inner product spaces.

A property of determinants is that if two rows (or columns) of a matrix are identical, the determinant is zero. This is because swapping two identical rows should change the sign of the determinant but also leaves the matrix unchanged, implying the determinant must be zero.

The kernel of a linear transformation T: V → W, denoted as ker(T), is the set of all vectors in the domain V that are mapped to the zero vector in the codomain W. In set notation, ker(T) = {v ∈ V | T(v) = 0}. The image (or range) is described in option 1.

The orthogonal projection of a vector v onto a subspace W is the vector in W that is closest to v. It's the 'shadow' of v onto W, and it is orthogonal to the component of v that is perpendicular to W. Options 2, 3, and 4 are incorrect interpretations of orthogonal projection.

Gaussian elimination is a method used to solve systems of linear equations. It transforms the system into an equivalent system in row-echelon form or reduced row-echelon form, making it easier to solve by back-substitution. While Gaussian elimination can also be used to compute determinants and find matrix inverses, its primary application is solving linear systems.

A linear transformation T must satisfy T(u + v) = T(u) + T(v) and T(cu) = cT(u). T(x, y) = (x + y, x - y) satisfies these conditions. The others fail: T(x, y) = (x^2, y) is not linear because of the x^2 term; T(x, y) = (x + 1, y) fails additivity due to the constant 1; T(x, y) = (xy, x) is not linear due to the xy term.

Row operations preserve the row space, rank, and null space (and thus the solution set of Ax=0) of a matrix. However, row operations change the column space of a matrix. While the row space is preserved, the actual columns themselves are transformed, thus changing the column space.

A square matrix A that satisfies the equation A^2 = A is called a projection matrix or idempotent matrix. Projection matrices are used to project vectors onto subspaces. Invertible matrices satisfy AA^(-1)=I, orthogonal matrices satisfy AA^T=I, and nilpotent matrices satisfy A^k=0 for some positive integer k.

An orthogonal matrix Q is a square matrix whose columns and rows are orthonormal vectors (unit vectors that are mutually perpendicular). A key property of orthogonal matrices is that their inverse is equal to their transpose, i.e., Q^(-1) = Q^T. Their determinant is ±1, their eigenvalues can be complex (though unitary), and their columns are linearly independent (and orthonormal).

A square matrix A is invertible (or non-singular) if there exists a matrix B such that AB = BA = I, where I is the identity matrix. Matrix B is called the inverse of A, denoted as A^(-1). An invertible matrix must have a non-zero determinant and none of its eigenvalues need to be zero (in fact, if 0 is an eigenvalue, it's not invertible). Being square is a necessary but not sufficient condition.

The dimension of the column space of a matrix A is defined as the rank of the matrix. The column space is the subspace spanned by the columns of A, and its dimension, the rank, represents the number of linearly independent columns (or rows) of the matrix. Nullity is the dimension of the null space.

A matrix is diagonalizable if it is similar to a diagonal matrix. Symmetric matrices are always diagonalizable over the real numbers. Singular or non-singular matrices are not necessarily diagonalizable (consider a shear matrix). Skew-symmetric matrices are not generally diagonalizable over the real numbers, although they are over the complex numbers.

If matrices A and A' represent the same linear transformation T but with respect to different bases, then A and A' are similar matrices. Similarity means there exists an invertible matrix P such that A' = P^(-1)AP, where P is the change-of-basis matrix from B' to B.

The trace of a square matrix is defined as the sum of its diagonal elements. It is also a fundamental property that the trace of a matrix is equal to the sum of its eigenvalues (counted with algebraic multiplicity). While the trace is also the sum of diagonal elements, the more conceptually significant answer in the context of linear algebra is its relation to eigenvalues.

By the Rank-Nullity Theorem, for a matrix A with n columns, rank(A) + nullity(A) = n, where nullity(A) is the dimension of the null space. Here, A has 4 columns (n=4), and nullity(A) = 2. Therefore, rank(A) = 4 - 2 = 2.

The inverse of the product of two invertible matrices AB is given by (AB)^(-1) = B^(-1)A^(-1). This is because (B^(-1)A^(-1))(AB) = B^(-1)(A^(-1)A)B = B^(-1)IB = B^(-1)B = I, and similarly (AB)(B^(-1)A^(-1)) = I. The order of inversion is reversed.

The characteristic polynomial of a square matrix A is defined as p(λ) = det(A - λI), where I is the identity matrix and λ is a scalar variable. The roots of the characteristic polynomial are the eigenvalues of the matrix A. Setting det(A - λI) = 0 gives the characteristic equation, whose solutions are the eigenvalues. While determinant and trace are related to eigenvalues, the primary purpose of the characteristic polynomial is to find eigenvalues.

A fundamental property of symmetric matrices is that eigenvectors corresponding to distinct eigenvalues are orthogonal. This is a key result in spectral theory and has important applications in various fields.

An eigenvalue λ of a matrix A is defined as a scalar such that there exists a non-zero vector v (eigenvector) satisfying Av = λv. This equation can be rewritten as (A - λI)v = 0. For a non-zero solution v to exist, the matrix (A - λI) must be singular, which means its determinant must be zero: det(A - λI) = 0. This is the characteristic equation.

Vector space axioms require closure under addition and scalar multiplication, and the existence of an additive identity. The existence of a multiplicative inverse for every non-zero element is a field axiom, not a vector space axiom directly related to the vector space structure itself, though the scalars are from a field.

A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. In option 1, (2, 0, 0) is a multiple of (1, 0, 0). In option 2, (1, 2, 1) = (1, 1, 0) + (0, 1, 1). In option 4, (2, 4, 6) = 2*(1, 2, 3). Only option 3, the standard basis vectors {(1, 0, 0), (0, 1, 0), (0, 0, 1)}, is linearly independent.