linear algebra questions True or False

i asked chat to make me some questions so i can practice

A set containing a single nonzero vector is always linearly independent.

True

If a set of vectors is linearly independent, then none of the vectors can be written as a linear combination of the others.

True

If the determinant of a square matrix is zero, its columns are linearly independent.

False

If the zero vector is in a set of vectors, the set is linearly dependent.

True

Two vectors are orthogonal if their dot product is zero.

True

An orthogonal matrix must have orthogonal columns of unit length(orthonormal)

True

All orthogonal matrices are square and invertible.

True

The inverse of an orthogonal matrix is equal to its transpose.

True

If two columns of a matrix are orthogonal, the matrix must be orthogonal.

False

A matrix is diagonalizable if and only if its Jordan form is diagonal.

True

The Jordan canonical form of a matrix is unique up to the order of Jordan blocks.

True

Every matrix has a Jordan form over the real numbers.

False

Jordan blocks are associated with distinct eigenvectors.

False

A Jordan block for eigenvalue λ has λ on the diagonal and 1’s on the superdiagonal.

True

Similar matrices have the same determinant and trace.

True

If A ∼ B, then A = PBP⁻¹ for some matrix P.

True

Similar matrices must have the same eigenvectors.

False

Diagonalizable matrices are similar to diagonal matrices.

True

Similar matrices always have the same Jordan canonical form.

True

Singular values are always non-negative real numbers.

True

The singular values of a matrix are the square roots of the eigenvalues of AᵀA.

True

A matrix with rank r has exactly r nonzero singular values.

True

The columns of V in the SVD A = UΣVᵀ are eigenvectors of AAᵀ.

False

Every square matrix has a singular value decomposition.

True

In QR factorization, Q is always an orthogonal matrix.

True

QR factorization can be used to solve least squares problems.

True

For a square matrix A, if A = QR, then R must be lower triangular.

False

Gram-Schmidt orthogonalization is one method to compute QR factorization.

True

The matrix Q in QR factorization has orthonormal columns.

True