Abstract Linear Algebra
Beginning from scratch and developing the standard topics of Linear Algebra, this book is intended as a text for a first course on the subject. The goal to which this work leads is the Theorem of Hurwitz - that the only normed algebras over the real numbers are the real numbers, the complex numbers, the quaternions, and the octonions. Unique in presenting this material at an elementary level, the book stresses the complete logical development of the subject and will provide a bavuable reference for mathematicians in general.
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A C Mn(k 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 example Exercise field finite-dimensional vector space function Given Hermitian Hint idempotent implies inner product space internal direct sum inverse isomorphism Jordan form Laplace expansion Lemma Lie Algebra linear combination linear map linearly independent minimal polynomial Mn(C Mn(R monic n x n matrix nonsingular nonzero normed algebra normed vector space Note orthogonal orthonormal basis positive definite Proof Proposition 12 prove real numbers real vector space ring satisfies scalar Schwarz Inequality Show similar matrices span standard basis subalgebra subset subspace Suppose surjective symmetric unique unit element unit vector unitary unitary matrix write xnvn zero