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 all the 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 awarded 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, actual Jordan, and rational canonical types. Covers normed linear vector areas, together with Banach areas. Discusses product areas, overlaying actual 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
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). Consequently, B is a basis of V. 9: Let V = V1 x be a finite product of vector spaces. , 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.
38: Let V and W be vector space over F, and let T e Hom(V, W). Then C:0-÷kerT V TIT >0 is an exact chain complex. Here i denotes the inclusion of ker T into V. 38 is a short exact sequence with V2 = ker T, d2 = i, V1 = V, etc. 40 is a short exact sequence if and only if d2 is injective, d1 is surjective, and Im d2 = ker d1. 33(c) implies that if C is a short exact sequence, then dim V2 — dim V1 + dim V0 = 0. 41: Suppose d, d, > > V0—÷0 is an exact chain complex. 0 (—1)' dim V1 = 0. 33(c) to each C1 and add the results, we get 3 (1) Let V and W be vector spaces over F.
Again we observe that if e &, then xx1 = xz. Thus (X, &) —* xx is a well-defined function from F x V/W to V/W. 4. Thus, (V/W,(&, fi) -+ & + fi, (x, &) -÷ x&) is a vector space over F. We shall refer to this vector space in the future as simply V/W. 12 imply that the natural map H: V V/W given by H(z) = & is a linear transformation. Clearly, H is surjective and has kernel W. 14: Suppose V is a finite-dimensional vector space over F and W a subspace of V. Then dim V = dim W + dim V/W. S We shall finish this section on quotients with three theorems that are collectively known as the isomorphism theorems.