Linear Algebra |
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a₁ a₂ affine arbitrary assume b₁ called characteristic polynomial commutative complex consider Corollary corresponding cyclic decomposition defined denote differential dimension direct sum divisor E₁ E₂ eigenspaces eigenvalues eigenvectors element exists F₁ fact finite follows formula ƒ and g G-graded g₁ given Hence homomorphism ideal implies induced inner product space irreducible subspaces Lemma Let f light-vector linear automorphisms linear isomorphism linear mapping linear space linear transformation linearly independent matrix minimum polynomial multiple n-dimensional N₁ nilpotent non-degenerate normed determinant function obtain orientation orthogonal complement orthonormal basis plane positive definite problem Proof proper rotation Proposition Prove pseudo-Euclidean quadric relation relatively prime respect satisfies selfadjoint semisimple skew mapping stable subspace subalgebra Suppose surjective symmetric bilinear function Theorem unit vector unitary space vector space vector xe whence x₁ y₁ yields zero