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Vector Spaces and Linear Maps
Matrices and Determinants
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A-stable algebra with unit associative bilinear form called characteristic polynomial choose a basis coefficients column commutative Corollary coset define Definition Let denote diagonal diagonalizable dimension direct sum e2 A e3 eigenvalue eigenvector End(V endomorphism equations equivalence classes example Exercise field finite-dimensional vector space function Given Hermitian Hint implies inner product space internal direct sum inverse isomorphism ith row Jordan form Laplace expansion Lemma Lie Algebra linear combination linear map linearly independent minimal polynomial Mn(C Mn(k Mn(R monic n x n matrix nonsingular nonzero normed algebra normed vector space Note orthogonal orthonormal basis positive definite Proof Proposition 9 prove real numbers real vector space ring satisfies scalar Schwarz Inequality Show similar matrices span stable subspace standard basis subalgebra subset Suppose surjective symmetric unique unit element unit vector unitary write zero