Đề thi, bài tập trắc nghiệm online Đại số tuyến tính – Đề 3

By definition, λ is an eigenvalue of A if and only if there exists a non-zero vector v such that Av = λv. This can be rewritten as Av - λv = 0, or (A - λI)v = 0. For this equation to have a non-zero solution v, the matrix (A - λI) must be singular, which means its determinant must be zero. If det(A - λI) were non-zero, (A-λI) would be invertible, and the only solution would be v=0, contradicting the definition of an eigenvector.

The Gram-Schmidt process is a method for orthonormalizing a set of vectors in an inner product space. Starting with a basis, it constructs an orthonormal basis that spans the same subspace. It does not diagonalize matrices, find eigenvalues/eigenvectors, or directly solve systems of equations, although orthonormal bases can be helpful in those contexts.

A crucial property of real symmetric matrices is that all their eigenvalues are real numbers. While eigenvectors can be complex in general matrices, real symmetric matrices guarantee real eigenvalues, which is important in many applications, especially in physics and engineering.

For a subset to be a subspace, it must be closed under addition and scalar multiplication, and contain the zero vector. The set of vectors of the form (x, 1) does not contain the zero vector (0, 0), and is not closed under scalar multiplication (e.g., 2*(x,1) = (2x, 2), which is not in the form (x, 1) unless x=1). The other options satisfy the subspace conditions.

To find the matrix representation, we consider how T transforms the standard basis vectors e₁ = (1, 0) and e₂ = (0, 1). Rotating e₁ by 90 degrees counterclockwise gives (0, 1), and rotating e₂ by 90 degrees counterclockwise gives (-1, 0). Placing these as columns forms the matrix [[0, -1], [1, 0]].

When you multiply a matrix A of size n x n by a scalar c, each of the n rows is multiplied by c. Since the determinant is multilinear in the rows (or columns), multiplying each of the n rows by c results in the determinant being multiplied by cⁿ. Therefore, det(cA) = cⁿdet(A).

A linear transformation is a function T: V → W between two vector spaces V and W that satisfies two key properties: additivity (T(u + v) = T(u) + T(v)) and homogeneity (T(cv) = cT(v)) for all vectors u, v in V and scalar c. These properties ensure that the linear structure of vector spaces is preserved under the transformation.

The vector space of all m x n matrices consists of matrices with m rows and n columns. A basis for this space can be formed by matrices with a single '1' in one position and '0' elsewhere (analogous to standard basis vectors in Rⁿ). There are m*n such positions, so the dimension of the space is m*n.

The determinant of a product of matrices is the product of their determinants, the determinant of the transpose is the same as the determinant of the original matrix, and scaling a matrix by k multiplies the determinant by kⁿ (for n x n matrices). However, in general, det(A + B) is NOT equal to det(A) + det(B). This is a common misconception.

According to the Rouché–Capelli theorem (also known as the Rank-Nullity Theorem in some contexts), a system of linear equations Ax = b has at least one solution if and only if the rank of the coefficient matrix A is equal to the rank of the augmented matrix [A|b]. If the ranks are different, the system is inconsistent and has no solution.

The inverse of the transpose of a matrix is equal to the transpose of the inverse of the matrix. That is, (Aᵀ)⁻¹ = (A⁻¹)ᵀ. This is a useful property when dealing with transposes and inverses in matrix algebra.

A square matrix is invertible (non-singular) if and only if several equivalent conditions are met. Having a null space containing only the zero vector is one of these conditions. This means that Ax = 0 only has the trivial solution x = 0. Other equivalent conditions include a non-zero determinant, full rank, and linearly independent columns (and rows).

Vector spaces are defined by closure under vector addition and scalar multiplication, along with other axioms like associativity and distributivity for these operations, and the existence of a zero vector and additive inverses. Closure under matrix multiplication is NOT a general property of vector spaces. Matrix multiplication is defined between matrices, and while a vector space can be formed by matrices of a certain size, the vector space itself isn't closed under matrix multiplication in the way it's closed under scalar multiplication and vector addition.

A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others. Option 4, {(1, 2, 3), (0, 1, 2), (0, 0, 1)}, is linearly independent because these vectors are not scalar multiples of each other, and no linear combination of two can produce the third. Option 1 is linearly dependent because (2,0,0) is a scalar multiple of (1,0,0). Option 2 is linearly dependent because (1,2,1) = (1,1,0) + (0,1,1). Option 3 contains more vectors than the dimension of the space, and is thus linearly dependent.

The dot product of two vectors is also known as the scalar product because the result of the operation is a scalar (a single number), not a vector. In contrast, the cross product (in 3D) results in a vector, and matrix product is between matrices or a matrix and a vector resulting in a matrix or vector respectively.

Two vectors are orthogonal if their dot product is zero. The dot product of u and v is (1)(3) + (2)(k) = 3 + 2k. Setting this to zero gives 3 + 2k = 0, so 2k = -3, and k = -3/2.

If A = A⁻¹, multiplying both sides by A gives A * A = A * A⁻¹, which simplifies to A² = I, where I is the identity matrix. Matrices that satisfy A² = I are called involutory matrices. While the identity matrix satisfies this, it's not the only case. For example, reflection matrices also satisfy A² = I and are their own inverses.

A matrix with a determinant of zero is singular, meaning it is not invertible. A non-invertible matrix does not have full rank, and the homogeneous system Ax = 0 has non-trivial solutions, thus infinitely many solutions. If the determinant were non-zero, then only the trivial solution would exist.

The column space of a matrix A, denoted as Col(A), is the span of the columns of A. This means it is the set of all possible linear combinations of the column vectors of A. Option 1 is generally false (row space and column space are not necessarily equal, though they have the same dimension - the rank). Option 2 is not generally true. Option 4 relates to the null space of Aᵀ, not A.

Matrix multiplication is associative, distributive over addition, and compatible with scalar multiplication. However, matrix multiplication is generally NOT commutative, meaning AB is not necessarily equal to BA. Commutativity holds only in special cases.

If a matrix has two identical rows (or columns), its determinant is always zero. This is a property of determinants and can be shown using row operations. Swapping the identical rows should change the sign of the determinant, but since the rows are identical, the matrix remains the same, implying the determinant must be zero.

The rank of a matrix is defined as the dimension of its column space (the space spanned by its column vectors), which is equal to the dimension of its row space (the space spanned by its row vectors). It represents the number of linearly independent rows or columns in the matrix.

An eigenvector v of a matrix A is a non-zero vector such that when A is multiplied by v, the result is a scalar multiple of v, i.e., Av = λv, where λ is the eigenvalue. This means the direction of the eigenvector remains unchanged (or reversed), only its magnitude is scaled by λ.

Gaussian elimination involves elementary row operations to transform a system of linear equations into row-echelon form. Multiplying two rows together is NOT a valid elementary row operation. The valid operations are swapping rows, scaling a row, and adding a multiple of one row to another.

The characteristic polynomial of a square matrix A is defined as det(A - λI), where λ is a scalar and I is the identity matrix. The roots of this polynomial are the eigenvalues of the matrix A. By solving the characteristic equation det(A - λI) = 0, we find the eigenvalues.

The cofactor Cᵢⱼ of an element aᵢⱼ in a square matrix A is defined as Cᵢⱼ = (-1)ⁱ⁺ʲMᵢⱼ, where Mᵢⱼ is the minor, which is the determinant of the submatrix obtained by deleting the i-th row and j-th column of A. The cofactor is used in calculating the determinant of a matrix and in finding the adjugate matrix.

Swapping two rows of a matrix changes the sign of its determinant. Since the identity matrix has a determinant of 1, swapping two rows results in an elementary matrix with a determinant of -1. Elementary matrices from row scaling have determinants equal to the scaling factor, and those from row addition have determinants of 1.

The null space (or kernel) of a matrix A is the set of all vectors x that, when multiplied by A, result in the zero vector. Mathematically, Null(A) = {x | Ax = 0}. Option 1 describes the column space, Option 3 describes the column space, and Option 4 describes the row space.

A matrix A is diagonalizable if it is similar to a diagonal matrix D, meaning there exists an invertible matrix P such that A = PDP⁻¹. Diagonalizable matrices are important because they simplify calculations and analysis, especially involving powers of matrices.

The trace of a square matrix is defined as the sum of the elements on the main diagonal (from the upper left to the lower right). It's a scalar value that provides some information about the matrix, but it's different from the determinant or the sum of all elements.