By William C. Brown

ISBN-10: 0471626023

ISBN-13: 9780471626022

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 issues supply the entire must haves 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 issues in linear algebra, the maths being provided 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 varieties of matrices, together with the Jordan, genuine Jordan, and rational canonical kinds. Covers normed linear vector areas, together with Banach areas. Discusses product areas, masking genuine internal product areas, self-adjoint changes, complicated internal product areas, and basic operators.

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

**Example text**

14) Suppose V is a finite-dimensional vector space over F, and let show there exists a polynomial f(X)e F[X] 5, T e HomF(V, V). If ST = such that S = f(T). 27 to prove the following theorem: Let V, W, Z be finite-dimensional vector spaces of dimensions n, m, and p, respectively. Let at, /1, and y be bases of V, W, and Z. If T e Hom(V, W) and Sc Hom(W, Z), then F(cx, y)(ST) = flfl, y)(S)f(at, fl)(T). (16) Suppose ->V1 d1 d1 tV0_+O and d'1 exists C' exact. Let T0 e HOmF(VO, V'Q). Show that there The for all i = 1 e HomF(VI, V) such that T1 - 1d1 = collection of linear transformations {T1} is called a chain map from C to C,.

These theorems appear in various forms all over mathematics and are very useful. 15 (First Isomorphism Theorem): Let T e HomF(V, V'), and suppose W is a subspace of V for which T(W) = 0. Let fl: V —* V/W be the natural map. 16: v/w Proof? We define I by T(&) = T(cz). Again, we remaind the reader that & is a subset of V containing cx. To ensure that our definition of I makes sense, we If e&, then; — oceW. Since T must argue that T(cz1) = T(x) for any is zero on W, we get T(x1) = Thus, our definition of I(&) depends only on the coset & and not on any particular representative of a.

16(b) implies that each P1 is a linear transformation such that Im P1 = V1. , P,,. , V,,. 16(b). 18: Let V = ent subspaces V1,. , projection maps. , P,, e Hom(V, V) are the associated (a) P1 = the identity map on V. 18(c)] in the algebra of endomorphisms t(V) = HomF(V, V). Let us take this opportunity to define some of the words in our last sentence. 19: By an associative algebra A over F, we shall mean a vector space /3) —* (A, + /3, (x, —÷ xcx) over F together with a second function (x, /1) —÷ cxfl from A x A to A satisfying the following axioms: Al.

### A Second Course in Linear Algebra by William C. Brown

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