By William C. Brown

ISBN-10: 0471626023

ISBN-13: 9780471626022

This textbook for senior undergraduate and primary yr graduate-level classes in linear algebra and research, covers linear algebra, multilinear algebra, canonical types of matrices, general linear vector areas and internal product areas. those issues supply all the must haves for graduate scholars in arithmetic to organize for advanced-level paintings in such components as algebra, research, topology and utilized mathematics.
Presents a proper method of complex issues in linear algebra, the math being awarded essentially via theorems and proofs. Covers multilinear algebra, together with tensor items and their functorial houses. Discusses minimum and attribute polynomials, eigenvalues and eigenvectors, canonical different types of matrices, together with the Jordan, genuine Jordan, and rational canonical varieties. Covers normed linear vector areas, together with Banach areas. Discusses product areas, protecting genuine internal product areas, self-adjoint changes, advanced internal product areas, and general operators.

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A function for some Te HomF(V, V') f: V -÷ V' is called an affine transformation if f = and some cc cV'. The set of all affine transformations from V to V' will be denoted AffF(V, V'). 10: If Ac d(V) and fe V'), then f(A) e d(V'). E Let us now return to the special subset V/W of d(V). The cosets of W can be given the structure of a vector space. 11, cc and /3 are vectors in V and & and /3 are their corresponding equivalence classes. & -I- if is defined to be the equivalence class that contains cc + /3.

20: F" a)' is a commutative diagram of isomorphisms. 20 is commutative. 15. We need only argue M(b, a) is an -÷ F" is an isomorphism. invertible matrix. Then the map M(b, a)t = SM(La): Now change of basis matrices M(ö, a) are always invertible. 18. For any fJeV, we have M(a, b)M(ö, a)[fl]5 = M(a, b)[fl]a = [fl]5. This equation easily implies M(a, ö)M(ö, a) = I,,, the n x n identity matrix. 6. 21: In this example, we construct a vector space isomorphic to V". Let V be a vector space over F, and let n e N.

Cx(/ly) = for all x, /3, yeA. A2. A4. x(xfl) =(xcx)/3 = x(xfl) for all fleA, xc F. A5. There exists an element I cA such that hx = = for all cxeA. We have seen several examples of (associative) algebras in this book already. and F[X] with the usual Any field F is an associative alebra over F. multiplication of matrices or polynomials is an algebra over F. If V is any vector space over F, then HomF(V, V) becomes an (associative) algebra over F when we define the product of two linear transformations T1 and T2 to be their composite T1T2.

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A Second Course in Linear Algebra by William C. Brown

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