Read e-book online A Second Course in Linear Algebra PDF

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 different types of matrices, basic linear vector areas and internal product areas. those themes offer 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 complex themes in linear algebra, the maths being offered essentially through 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, actual Jordan, and rational canonical varieties. Covers normed linear vector areas, together with Banach areas. Discusses product areas, masking genuine internal product areas, self-adjoint variations, advanced internal product areas, and basic operators.

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11k Let A and B be subsets of V (or V*), (a) A c B implies A' (b) L(A)' = A'. B'. (c) (AuB)'=A'nB'. (d) A c A". (e) If W is a subspace of a finite-dimensional vector space V, then dimV = dimW + dimW1. Proof (a)—(d) are straightforward, and we leave their proofs as exercises. We prove (e). Let .. , be a basis of W. , of V. Thus, dimW = m and dimv = n. Let be a dual basis of We complete the proof of (e) by arguing is a basis of W'. ,m. In particular, Since c is linearly independent over F. We must show L({c4+1,.

TON). Again the reader can easily check that 'F is an injective, linear transformation. 7 implies that 'F is surjective. fl PRODUCTS AND DIRECT SUMS 33 x Suppose V = V1 x is a finite product of vector spaces over F. , n. We can think of the vectors in V as n-tuples with oc1eV1. ,O). Thus, 91(x) is the n-tuple of V that is zero everywhere except for an in the ith slot. Since °i is injective, 01: V1 01(V1). Since 01(B1) n 01(V1). In particular, 01(B1) is a basis of the subspace = (0), B = 91(B1) is a linearly independ- ent set Clearly, V = >J' 101(V1).

9) Find three subspaces V= V1, that V V V such that WcV2 and V=V1EJ3W. (11) Let A be an algebra over F. A linear transformation TeHomF(A, A) is called an algebra homomorphism if T(xfl) = TOx)T(fl) for all fleA. Ex- hibit a nontrivial algebra homomorphism on the algebras F[X] and 38 LINEAR ALGEBRA (12) Suppose V is a vector space over F. Let 5: Show that the map T V be an isomorphism of V. - 'TS is an algebra homomorphism of t(V) which is one to one and onto. S (13) Let F be a field. Show that the vector space V = F (over F) is not the direct sum of any two proper subspaces.

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A Second Course in Linear Algebra by William C. Brown

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