By William C. Brown

This textbook for senior undergraduate and primary yr graduate-level classes in linear algebra and research, covers linear algebra, multilinear algebra, canonical different types of matrices, common linear vector areas and internal product areas. those themes offer all the must haves for graduate scholars in arithmetic to arrange for advanced-level paintings in such components as algebra, research, topology and utilized mathematics.

Presents a proper method of complicated subject matters in linear algebra, the math being provided basically by way of theorems and proofs. Covers multilinear algebra, together with tensor items and their functorial houses. Discusses minimum and attribute polynomials, eigenvalues and eigenvectors, canonical kinds of matrices, together with the Jordan, genuine Jordan, and rational canonical types. Covers normed linear vector areas, together with Banach areas. Discusses product areas, masking actual internal product areas, self-adjoint changes, complicated internal product areas, and basic operators.

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We call w a bilinear map if for all cV, w(; ) e HomF(V', W) and for all ,6e V', co(, fJ)e HomF(V, W). Thus, a function V x V' -÷ W is a /1) + yw(cx2, /3), and ocr, cx2eV and /3, co: bilinear map if and only if + w(xcz1 + fl2eV'. 6, e V and T e V*. The fact that w is a bilinear map is obvious. w determines a natural, injective, linear transformation i/i: V —. V** eV, set in the following way. If = T) = ye F, ç&(z) = w(cx, ). /3 e V Thus, for any T e and T e V*, then + yfl)(T) = w(xcz + y/J, T) = xco(cz, T) + yw(fl, T) = (xI/ar(z) + Consequently, if, e HomF(V, V**).

If is a basis of V, then any T e Hom(V, V) is represented in terms of by an n x n matrix A = cx)(T). If we change to a of V, then the representation of T changes to B = JT(cx', cx')(T). 29 implies that B = PAP 1, where P = M(a, a'). Recall that two n x n matrices A and B are similar if there exists an invertible n x n matrix P Thus, different representations of the same such that B = T e Hom(V, V) with respect tO different bases of V are similar matrices. Now we can ask, What is the simplest representation of T?

Im T is called the image of T and is a subspace of W. Before proceeding further, let us give a couple of important examples of isomorphisms between vector spaces. 12: M1 via the transpose A -÷ At. We have already = M1 JF). Thus, all three of the vector spaces M1 JF), and F are isomorphic to each other. 13: Suppose V is a finite-dimensional vector space over F. Then every basis = {x1,. , = /1. T(oc) is just the 1(F) and the transpose composite of the coordinate map V —' 1(F) —÷ M1 JF) = F. Since both of these maps are isomorphisms, we see is an isomorphism.