The eigenvalues of a scalar multiple of a matrix are the scalar multiples of the eigenvalues.
The eigenvalues of a triangular matrix are the entries on the main diagonal.
The eigenvalues of the inverse of a nonsingular matrix are the reciprocals of the eigenvalues.
The eigenvectors of a normal matrix are an orthonormal basis.
The geometry of linear systems
The Gram-Schmidt process converts a linearly independent set into an orthogonal set.
The identity matrix is the identity for matrix multiplication.
The image of a linearly dependent set under a linear transformation is linearly dependent.
The image of a linearly independent set under an injective linear transformation is linearly independent.
The inner product of a vector with itself is the square of its norm/length.
The intersection of subspaces is a subspace
The inverse image of a subspace under a linear transformation is a subspace.
The inverse of a linear transformation is a linear transformation
The inverse of a matrix can be expressed in terms of its matrix of cofactors.
The inverse of a matrix can be used to solve a linear system.
The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix.
The inverse of an invertible upper/lower triangular matrix is upper/lower triangular.
The inverse of an isomorphism is an isomorphism.
The inverse of a scalar multiple is the reciprocal times the inverse.
The inverse of the inverse of a linear transformation is the original linear transformation
The kernel/null space of a linear transformation is a subspace
The kernels of powers of a linear transformation form an ascending chain
The least squares solution to a linear system is unique if and only if the columns of the coefficient matrix are linearly independent.
The left null space of a matrix is a subspace of R^m (or C^m).
The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A.

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