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 different types of matrices, basic linear vector areas and internal product areas. those subject matters supply the entire necessities for graduate scholars in arithmetic to organize for advanced-level paintings in such parts as algebra, research, topology and utilized mathematics.
Presents a proper method of complex themes in linear algebra, the maths being awarded essentially via theorems and proofs. Covers multilinear algebra, together with tensor items and their functorial homes. Discusses minimum and attribute polynomials, eigenvalues and eigenvectors, canonical sorts of matrices, together with the Jordan, genuine Jordan, and rational canonical kinds. Covers normed linear vector areas, together with Banach areas. Discusses product areas, protecting genuine internal product areas, self-adjoint alterations, complicated internal product areas, and common operators.
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Extra info for A Second Course in Linear Algebra
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. 10: For any indexing set A, Hom(W, flEa HIGA Hom(W, V1).
Set K = ker T. Show there exists a one-to-one, inclusion-preserving correspondence between the subspaces of V' and the subspaces of V containing K. (6) Let Te Hom(V, V'), and let K = ker T. Show that all vectors of V that have the same image under T belong to the same coset of V/K. (7) Suppose W is a finite-dimensional subspace of V such that V/W is finite dimensional. Show V must be finite dimensional. 45 EXERCISES FOR SECTION 5 (8) Let V be a finite-dimensional vector space. If W is a subspace with dim W = dim V — 1, then the cosets of W are called hyperplanes in 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. 4: Let V = F, that is, V is the direct sum of the vector spaces follows from Exercise 2 of Section 4 that V* = e N}. It = Fli F, F) flit' HomF(F, F) F. 13, we know that dim V = NI.