Finite-dimensional Vector Spaces

adjoint algebraic multiplicity arbitrary assertion bilinear commutes complex numbers complex vector space concept conjugate consider converse coordinate system correspondence coset defined definition denote diagonal dimension direct sum disjoint dual space elements equation equivalent exists fact finite finite-dimensional inner product finite-dimensional unitary space finite-dimensional vector space follows Hilbert space implies inner product space invariant invertible isometry isomorphism k-linear form linear combination linear functional linear transformation linearly independent M₁ matrix multilinear n-dimensional vector space necessary and sufficient nilpotent non-zero normal transformations notation observe obtain orthogonal orthonormal basis partial isometry permutation perpendicular projection polynomial proof proper value proper vector properties Prove real numbers real vector space relation respect result scalar self-adjoint transformation spectral theorem strictly positive subset sufficient condition Suppose tensor product theory tion unitary space write y₁ zero