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

The rank of a matrix is equal to the rank of its transpose. This is because the row rank and column rank of a matrix are always equal. Options 2, 3, and 4 are not generally true. For instance, rank(A^2) can be less than rank(A), and rank(A) is always equal to rank(A^T), not greater or less.

A linear transformation T must satisfy two conditions: T(u + v) = T(u) + T(v) and T(cu) = cT(u). T(x) = x + 1 does not satisfy T(0) = 0, a necessary condition for linear transformations. T(0) = 0 + 1 = 1, not 0. The others (T(x) = 2x, T(x) = -x, T(x) = 0) are linear transformations.

A square matrix A is called idempotent (or a projector) if A^2 = A. Involutory matrices satisfy A^2 = I. Nilpotent matrices satisfy A^k = 0 for some positive integer k. Orthogonal matrices satisfy A^T * A = I.

The span of a set of vectors is defined as the set of all possible linear combinations of those vectors. This set forms a vector subspace. It is also the smallest subspace that contains all the given vectors.

When there are more unknowns than equations, the system is underdetermined. This means there are fewer constraints than variables, leading to either infinitely many solutions or no solution. It cannot have a unique solution or exactly two solutions in such a case.

Even if A and B are invertible, their sum (A + B) is not necessarily invertible. For example, consider A = I and B = -I, both invertible, but A + B = 0, which is not invertible. Invertibility of A and B does not guarantee invertibility of their sum.

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as (a*d) - (b*c). This is a fundamental formula for 2x2 determinants.

The dimension of the column space of a matrix A is called the rank of A. The rank represents the number of linearly independent columns (or rows) in the matrix, and thus the dimension of the space spanned by the columns.

Gaussian elimination (or row reduction) is generally the most efficient method for solving large systems of linear equations. Cramer`s rule is computationally expensive for large systems due to determinants. Finding the inverse and multiplying is also generally less efficient and numerically less stable than Gaussian elimination. Graphical methods are only suitable for very small systems (typically 2 variables).

A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others. {[1, 0], [0, 1]} is linearly independent because neither vector is a scalar multiple of the other. Option 1 is linearly dependent because [2, 4] = 2*[1, 2]. Option 2 has more vectors than the dimension of R^2, thus linearly dependent. Option 4 contains the zero vector, making it linearly dependent.

The kernel of a linear transformation T: V → W 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_W}.

By definition, a square matrix A is orthogonal if its transpose is equal to its inverse, i.e., A^T = A^(-1). Therefore, A^T * A = A^(-1) * A = I, the identity matrix.

Determinants have the property that det(AB) = det(A)det(B) and det(A^T) = det(A). Also, if a matrix has a row or column of zeros, its determinant is 0. However, det(A + B) is generally NOT equal to det(A) + det(B). This is a common misconception.

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. This is defined by the equation Ax = 0.

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).

According to the Rouché–Capelli theorem, a system of linear equations Ax = b is consistent (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]. Determinant condition relates to unique solution, not just consistency. A doesn`t have to be square for consistency. b being zero vector ensures consistency but is not a necessity for consistency in general.

An eigenvector v of a square matrix A is a non-zero vector that, when multiplied by A, results in a scalar multiple of itself. This is mathematically expressed as Av = λv, where λ is the corresponding eigenvalue.

A fundamental property of symmetric matrices is that eigenvectors corresponding to distinct eigenvalues are always orthogonal. This is a crucial result in linear algebra and has many applications.

Two vectors are orthogonal (perpendicular) if and only if their dot product is zero. This is a fundamental property of orthogonality in vector spaces.

A vector space must be closed under addition and scalar multiplication, and contain a zero vector. The set of solutions to a non-homogeneous linear differential equation does not contain the zero function (unless the non-homogeneous part is zero), and is not closed under scalar multiplication (if y is a solution, 2y is not necessarily a solution if the equation is non-homogeneous). Polynomials of degree ≤ 2, 2x2 matrices, and R^3 are vector spaces.

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

A square matrix is invertible if and only if its determinant is non-zero. If the determinant is 0, the matrix is singular, meaning it is not invertible.

If Av = λv, then (A + kI)v = Av + kIv = λv + kv = (λ + k)v. Therefore, if λ is an eigenvalue of A, then λ + k is an eigenvalue of A + kI. Adding kI to a matrix shifts its eigenvalues by k.

A determinant of -1 for a linear transformation in R^2 indicates a reflection. A determinant of 1 often corresponds to rotations or shears that preserve orientation. Scaling changes the magnitude of determinant but not necessarily the sign to negative one unless scaling factor is negative which is not standard geometric interpretation in this context. A determinant of -1 specifically signifies a reflection, which reverses the orientation.

A subspace must contain the zero vector and be closed under addition and scalar multiplication. The set of vectors (x, y, z) such that x - y = 0 (i.e., x = y) is a subspace because it contains the zero vector (0, 0, 0), and is closed under addition and scalar multiplication. Option 1 does not contain the zero vector. Option 2 is not closed under scalar multiplication by negative scalars. Option 4 is not closed under addition.

For a square matrix, the sum of its eigenvalues (counted with algebraic multiplicity) is equal to its trace, and the product of its eigenvalues is equal to its determinant. Option 4 correctly states the relationship between the sum of eigenvalues and the trace.

For an injective (one-to-one) linear transformation T: V → W, the dimension of the domain V must be less than or equal to the dimension of the codomain W. Injectivity implies that distinct vectors in V map to distinct vectors in W, which is only possible if W is `at least as large` as V in terms of dimension.

The characteristic polynomial of a matrix A is defined as det(A - λI) = 0, where λ represents eigenvalues and I is the identity matrix. Solving this equation for λ gives the eigenvalues of matrix A.

A square matrix of size n x n is diagonalizable if and only if it has n linearly independent eigenvectors. This set of eigenvectors can form a basis for R^n, allowing the matrix to be diagonalized. While symmetric matrices are always diagonalizable, invertibility or determinant value of 1 are not sufficient or necessary conditions for diagonalizability in general.

Gaussian elimination involves elementary row operations to transform a matrix into row-echelon form. The valid operations are swapping rows, scaling rows, and adding multiples of rows. Multiplying two rows is not a valid elementary row operation.