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- The eigenvalues of a polynomial of a matrix are the polynomial of the eigenvalues.#23691
- The eigenvalues of a power of a matrix are the power the eigenvalues.#23690
- The eigenvalues of a scalar multiple of a matrix are the scalar multiples of the eigenvalues.#23689
- The eigenvalues of a triangular matrix are the entries on the main diagonal.#23709
- The eigenvalues of the inverse of a nonsingular matrix are the reciprocals of the eigenvalues.#23692
- The eigenvectors of a normal matrix are an orthonormal basis.#23602
- The geometry of linear systems#23427
- The Gram-Schmidt process converts a linearly independent set into an orthogonal set.#23936
- The identity matrix is the identity for matrix multiplication.#23498
- The image of a linearly dependent set under a linear transformation is linearly dependent.#24065
- The image of a linearly independent set under an injective linear transformation is linearly independent.#24032
- The inner product of a vector with itself is the square of its norm/length.#23910
- The intersection of subspaces is a subspace#23967
- The inverse image of a subspace under a linear transformation is a subspace.#24057
- The inverse of a linear transformation is a linear transformation#24028
- The inverse of a matrix can be expressed in terms of its matrix of cofactors.#23721
- The inverse of a matrix can be used to solve a linear system.#23532
- The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix.#23527
- The inverse of an invertible upper/lower triangular matrix is upper/lower triangular.#23565
- The inverse of an isomorphism is an isomorphism.#24002
- The inverse of a scalar multiple is the reciprocal times the inverse.#23525
- The inverse of the inverse of a linear transformation is the original linear transformation#24029
- The kernel/null space of a linear transformation is a subspace#24059
- The kernels of powers of a linear transformation form an ascending chain#24062
- The least squares solution to a linear system is unique if and only if the columns of the coefficient matrix are linearly independent.#23950