By William C. Brown

This textbook for senior undergraduate and primary 12 months graduate-level classes in linear algebra and research, covers linear algebra, multilinear algebra, canonical kinds of matrices, basic linear vector areas and internal product areas. those issues supply the entire necessities 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 maths being offered basically by way of theorems and proofs. Covers multilinear algebra, together with tensor items and their functorial homes. Discusses minimum and attribute polynomials, eigenvalues and eigenvectors, canonical varieties 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 genuine internal product areas, self-adjoint modifications, advanced internal product areas, and basic operators.

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**Extra info for A Second Course in Linear Algebra**

**Example text**

Let is a basis of V*. Suppose We claim that z* = . , n}. 3 implies 0 = is linearly independent over F. = 0, and y1 = This last equation follows immediately If T e V*, then T = = V*, and is a basis of V*. 3. 3 is called the dual basis of of a finite-dimensional vector space V has a corresponding of V*. k e x*. If V is not finite dimensional over F, then the situation is quite different. 2 is false when dim V = cc. If dim V = cc, then dim V* > dim V. Instead of proving that fact, we shall content ourselves with an example.

Let T = and T1_1d1 = V11 x and C' = {(V, lie 71} are two chain be a chain map from C to C'. Thus, T1: V1 —÷ = set For each for all by Define a map di': id. id, 30 LINEAR ALGEBRA Show that C" = {(Vr, dflhiel} is a fi) = (—d1_1(x), T1_1(oc) + chain complex. The complex C" is called the mapping cone of T. 33(c) to give another proof of Exercise 16 in Section 2. (19) Find a T e C) that is not C-linear. (20) Let V be a finite-dimensional vector space over F. Suppose T e HomF(V, V) such that dim(Im(T2)) = dim(Im(T)).

N, then B = 01(B1) is a basis of V. In particular, if each V1 is finite dimensional, then so is V. In this case, we have dimV=ThidimV1. u At this point, let us say a few words about our last three theorems when Al = cc. 6 is true for any indexing set A. The map 'P(T) = (ir1T)166 is an injective, linear transformation as before. 5 to conclude 'I' is surjective, since 01T1 makes no sense when Al = cc. However, we can argue directly that 'P is surjective. Let (T1)ICA e Hom(W, V1). Define V1) by T(x) = Clearly 'I'(T) = (T1)IEA.