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- A linear transformation is surjective if and only if the rank equals the dimension of the codomain.#24035
- A linear transformation maps 0 to 0.#23864
- A linear transformation of a linear combination is the linear combination of the linear transformation#23865
- A linear transformation on a finite dimentional nontrivial vector space has at least one eigenvalue.#24049
- All echelon forms of a linear system have the same free variables#23423
- A matrix and its transpose have the same determinant.#23728
- A matrix and its transpose have the same eigenvalues/characteristic polynomial.#23693
- A matrix A with real entries has orthonormal columns if and only if A inverse equals A transpose.#23930
- A matrix equation is equivalent to a linear system#23767
- A matrix is called ill-conditioned if it is nearly singular#23749
- A matrix is nilpotent if and only if its only eigenvalue is 0.#23583
- A matrix is orthogonally diagonalizable if and only if it is normal (The principal axis theorem).#23601
- A matrix is orthogonally diagonalizable if and only if it is symmetric.#23578
- A matrix of rank k is equivalent to a matrix with 1 in the first k diagonal entries and 0 elsewhere.#23606
- A matrix turns into its adjoint when moved to the other side of the standard inner product on C^n.#23906
- A matrix with a 0 row/column has determinant 0#23741
- A matrix with real entries and orthonormal columns preserves dot products.#23932
- A matrix with real entries and orthonormal columns preserves norms.#23931
- A matrix with real entries has eigenvalues occurring in conjugate pairs.#23710
- A matrix with two equal rows/columns has determinant 0#23740
- An eigenspace of a matrix is a nontrivial subspace.#23685
- An eigenspace of a matrix is the null space of a related matrix.#23686
- An n-by-n matrix is diagonalizable if and only if it has n linearly independent eigenvectors.#23612
- An n-by-n matrix is diagonalizable if and only if the characteristic polynomial factors completely#23614
- An n-by-n matrix is diagonalizable if and only if the sum of the dimensions of the eigenspaces equals n.#23613
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