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- The dimension of a direct sum of subspaces is the sum of the dimensions of the subspaces.#23971
- The dimension of a eigenspace is less than or equal to the (algebraic) multiplicity of the eigenvalue.#23708
- The dimension of a subspace is less than or equal to the dimension of the whole space#24010
- The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.#24033
- The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.#24034
- The direct sum of a subspace and its orthogonal complement is the whole space.#23922
- The echelon form can be used to determine if a linear system is consistent.#23771
- The eigenspace of a linear transformation is a nontrivial subspace.#24051
- The eigenvalues of a matrix are the roots/solutions of its characteristic polynomial/equation.#23705
- 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 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 independent set under an injective linear transformation is linearly independent.#24032
- The image of a spanning set spans the range#24037
- 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 composition reverses the order#24030
- 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