Đề 6 – Bài tập, đề thi trắc nghiệm online Đại số tuyến tính

For large systems of linear equations, Gaussian elimination (or its variations like LU decomposition) is generally the most computationally efficient method. Cramer`s rule and finding the inverse matrix are computationally expensive for large matrices. The graphical method is only suitable for very small systems (typically 2 variables).

For a subset to be a subspace of a vector space, it must satisfy three conditions: it must contain the zero vector, be closed under scalar multiplication, and be closed under vector addition.

For matrix multiplication AB to be defined, the number of columns of matrix A must be equal to the number of rows of matrix B. The resulting matrix will have the number of rows of A and the number of columns of B.

A system with more equations than variables is called an overdetermined system. It may or may not have a solution. Conversely, a system with fewer equations than variables is underdetermined.

The determinant of the identity matrix (of any size) is always 1. This can be shown by cofactor expansion or by considering that the identity transformation scales volume by a factor of 1.

For a 2x2 matrix, the absolute value of its determinant represents the factor by which the linear transformation associated with the matrix scales areas. For 3x3 matrices, it represents the volume scaling factor, and in general, for nxn matrices, it`s the n-dimensional volume scaling factor.

The characteristic polynomial of a matrix A is defined as 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.

If the determinant of the coefficient matrix A is non-zero, it means A is invertible. In this case, the system Ax = b has a unique solution given by x = A^(-1)b. This is a consequence of Cramer`s rule and the Invertible Matrix Theorem.

For a square matrix, the sum of its eigenvalues is equal to the trace of the matrix (the sum of the diagonal elements). The product of eigenvalues is equal to the determinant.

The rank of a matrix is the dimension of its column space (which is always equal to the dimension of its row space). It represents the maximum number of linearly independent columns (or rows) in the matrix.

A square matrix is invertible if and only if its determinant is non-zero. If det(A) = 0, the matrix is singular or non-invertible.

A linear transformation T must satisfy two properties: T(u + v) = T(u) + T(v) and T(cu) = cT(u) for scalars c and vectors u, v. f(x) = 3x satisfies these properties. f(x) = x^2, f(x) = sin(x), and f(x) = 2x + 3 (due to the constant term) do not.

A basis for a vector space V is a set of vectors that is both linearly independent and spans V. A basis does *not* contain the zero vector (except in the trivial vector space {0}). It must be a subset of V.

For an invertible matrix A, the determinant of its inverse A^(-1) is the reciprocal of the determinant of A, i.e., det(A^(-1)) = 1/det(A). This follows from the property det(AB) = det(A)det(B) and det(AA^(-1)) = det(I) = 1.

A matrix A is diagonalizable if it is similar to a diagonal matrix D. This means there exists an invertible matrix P such that A = PDP^(-1). Diagonalizability depends on having a sufficient number of linearly independent eigenvectors.

Rotation, reflection, and scaling are linear transformations because they preserve vector addition and scalar multiplication through the origin. Translation (shifting by a constant vector) is not linear unless the translation vector is the zero vector, because it doesn`t preserve the origin (T(0) is not 0 unless the translation is zero).

The column space of a matrix A, denoted Col(A), is the span of the columns of A. It`s the set of all possible linear combinations of the column vectors of A.

The solution set to a non-homogeneous linear system Ax = b is not a vector space (unless b=0) because it does not contain the zero vector. It is an affine subspace, which is a translation of the null space of A. It`s formed by adding a particular solution to the homogeneous system Ax = 0 to each vector in the null space.

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}.

The Rank-Nullity Theorem states that for any matrix A with n columns, the sum of the dimension of the null space (nullity) and the dimension of the column space (rank) is equal to the number of columns, n. dim(Null(A)) + rank(A) = n.

Row equivalent matrices have the same null space. Row operations preserve the solution set of Ax = 0, hence preserving the null space. Determinants and eigenvalues are generally not preserved under row operations. The column space can change, but the row space remains the same (or related).

A key property of symmetric matrices (A = A^T) is that all their eigenvalues are real numbers. This is a fundamental result in linear algebra and has significant implications in various applications. Not all symmetric matrices are orthogonal or non-invertible; invertibility depends on whether zero is an eigenvalue.

Vectors are linearly independent if no vector in the set can be expressed as a linear combination of the remaining vectors. This means the only linear combination that equals the zero vector is the trivial combination (all coefficients are zero).

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

The dimension of the vector space R^n is n. R^3 is the set of all 3-tuples of real numbers, and its standard basis has 3 vectors: (1,0,0), (0,1,0), (0,0,1). Therefore, its dimension is 3.

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

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

The trace of a square matrix is the sum of the elements on its main diagonal (from the upper left to the lower right).

The Gram-Schmidt process is an algorithm used to orthogonalize a set of vectors in an inner product space. It takes a set of linearly independent vectors and produces an orthonormal basis for the subspace they span.

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 scalar multiple λ is called the eigenvalue corresponding to v. The defining equation is Av = λv.