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

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