Đề 12 – Bài tập, đề thi trắc nghiệm online Đại số tuyến tính

A square 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, where D is a diagonal matrix.

A subspace must contain the zero vector and be closed under addition and scalar multiplication. Option 3, the set of vectors (x, x, 0), is a subspace as it satisfies these conditions. Option 1 does not contain the zero vector. Option 2 is not closed under scalar multiplication. Option 4 is not closed under addition.

A 3x3 symmetric matrix has the form [[a, b, c], [b, d, e], [c, e, f]]. There are 6 independent entries (a, b, c, d, e, f) that determine the matrix. Therefore, the dimension of the vector space of 3x3 symmetric matrices is 6.

The rank-nullity theorem states that for any matrix A, rank(A) + nullity(A) = n, where n is the number of columns of A, rank(A) is the dimension of the column space, and nullity(A) is the dimension of the null space.

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

Determinants have the property that det(AB) = det(A)det(B) and det(A^T) = det(A). If a matrix has a row of zeros, its determinant is 0. However, in general, det(A + B) ≠ det(A) + det(B).

A basis for R^2 must consist of exactly two linearly independent vectors that span R^2. {(1, 0), (0, 1)} is the standard basis for R^2 and satisfies these conditions. Options 1 and 2 are linearly dependent. Option 3 has more than two vectors, so it cannot be a basis for R^2 (although it spans R^2).

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc.

If Av = λv, then A^2v = A(Av) = A(λv) = λ(Av) = λ(λv) = λ^2v. Thus, if λ is an eigenvalue of A, then λ^2 is an eigenvalue of A^2.

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 all elements in a row is not an elementary row operation because it is not reversible in the same way and does not preserve the solution set of a linear system.

A system with more equations than variables is called an overdetermined system. It may or may not have a solution, but it is characterized by having more constraints than degrees of freedom.

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 that spans the same subspace.

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

A square matrix A is called idempotent if A^2 = A. Idempotent matrices are often used in projection operations in linear algebra.

Matrix multiplication is associative, meaning (AB)C = A(BC). It is generally not commutative (AB ≠ BA), does not always result in a square matrix, and is only defined when the number of columns in the first matrix equals the number of rows in the second matrix.

A linear transformation T must satisfy T(u + v) = T(u) + T(v) and T(cu) = cT(u) for scalars c and vectors u, v. T(x, y) = (2x, 3y) satisfies these conditions. The others fail due to squaring, addition of a constant, or multiplication of variables.

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

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

If A is an nxn matrix and c is a scalar, then det(cA) = c^n det(A). In this case, n=3 and c=2, so det(2A) = 2^3 det(A) = 8 * 5 = 40.

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

Two vectors are orthogonal (perpendicular) if and only if their dot product is zero.

The null space of a matrix A, denoted Nul(A), is the set of all solutions to the homogeneous equation Ax = 0. It is a subspace of the domain of the linear transformation defined by A.

Vectors v1, v2, ..., vn are linearly dependent if there exist scalars c1, c2, ..., cn, not all zero, such that c1v1 + c2v2 + ... + cn vn = 0. This implies that at least one vector can be written as a linear combination of the others.

Geometrically, each linear equation in two variables represents a line in a plane. If a system of two linear equations has no solution, it means the corresponding lines do not intersect, which occurs when they are parallel and distinct.

The Invertible Matrix Theorem lists many equivalent conditions for a square matrix to be invertible. Diagonalizability is not a condition for invertibility. While many invertible matrices are diagonalizable, not all are required to be (and diagonalizability is a separate concept related to eigenvalues and eigenvectors).

The rank of a matrix is defined as the dimension of its column space (or equivalently, its row space). It represents the maximum number of linearly independent columns (or rows) in the matrix.

The column space of a matrix A, denoted Col(A), is the set of all linear combinations of the columns of A. It is the span of the columns of A and is a subspace of the codomain of the linear transformation defined by A.

The set of vectors in R^3 with non-negative components is not a vector space because it is not closed under scalar multiplication. For example, if you multiply a vector with positive components by a negative scalar, the resulting vector will have negative components, which is not in the set.

By the Spectral Theorem, every symmetric matrix is diagonalizable, and its eigenvectors corresponding to distinct eigenvalues are orthogonal. Invertible and singular matrices are not necessarily diagonalizable. Skew-symmetric matrices are not guaranteed to be diagonalizable over real numbers (though they are over complex numbers).

The vectors are linearly independent if and only if the determinant of the matrix formed by these vectors as columns is non-zero. The determinant of [[1, 0, 0], [2, 1, 0], [3, 2, k]] is 1*(1*k - 0*2) - 0 + 0 = k. The determinant is non-zero if k ≠ 0. Thus, the vectors are linearly independent for k ≠ 0.