Linear Algebra

adjoint B₁ c₁ characteristic polynomial characteristic vectors column complex numbers Corollary cyclic vector defined Definition degree denote diagonal matrix diagonalizable dimension direct sum E₁ element entries equations equivalence relation EXAMPLE Exercise field F field of complex finite-dimensional inner product finite-dimensional vector space follows hence inner product space invariant invertible isomorphism Jordan form lemma Let F linear combination linear functional linear operator linear transformation linearly independent matrix over F minimal polynomial monic n-linear nilpotent nomial non-degenerate non-negative non-zero vector normal operators null space obtain orthonormal basis p₁ permutation polynomial f polynomial over F positive integer primary decomposition theorem prime Prove rank real numbers row-equivalent scalar field scalar multiple self-adjoint Show standard inner product standard ordered basis subfield subset subspace spanned Suppose symmetric bilinear form T-annihilator T₁ unique V₁ vector ẞ vectors a₁ W₁ Xn matrix zero αι απ