By Steven H. Weintraub

Linear algebra occupies a vital position in glossy arithmetic. This e-book offers a rigorous and thorough improvement of linear algebra at a complicated point, and is directed at graduate scholars mathematicians. It ways linear algebra from an algebraic viewpoint, yet its collection of themes is ruled not just for his or her value in linear algebra itself, but additionally for his or her functions all through arithmetic. scholars in algebra, research, and topology will locate a lot of curiosity and use to them, and the cautious therapy and breadth of subject material will make this e-book a worthwhile reference for mathematicians all through their specialist lives.

Topics taken care of during this booklet contain: vector areas and linear variations; size counting and functions; illustration of linear adjustments by means of matrices; duality; determinants and their makes use of; rational and particularly Jordan canonical shape; bilinear kinds; internal product areas; common linear changes and the spectral theorem; and an advent to matrix teams as Lie groups.

The booklet treats vector areas in complete generality, although it concentrates at the finite dimensional case. additionally, it treats vector areas over arbitrary fields, specializing to algebraically closed fields or to the fields of genuine and complicated numbers as necessary.

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If m D n, choose a basis fv1 ; : : : ; vm g of V and a basis fw1; : : : ; wm g of W . 24 T is an isomorphism. ✐ ✐ ✐ ✐ ✐ ✐ “book” — 2011/3/4 — 17:06 — page 20 — #34 ✐ ✐ 20 1. 7. Let A be an n-by-n matrix. The following are equivalent: (1) A is invertible. (10 ) The equation Ax D b has a unique solution for every b 2 F n . (2) The equation Ax D b has a solution for every b 2 F n . (3) The equation Ax D 0 has only the trivial solution x D 0. Proof. 2 into matrix language. We emphasize that this one-sentence proof is the “right” proof of the equivalence of these properties.

2 to the linear transformation TA . AB/ D A 1 I D A 1 , and similarly if BA D I . 4. 2 is false in the infinite-dimensional case: (1) Let V D r F 11 and consider left shift L and right shift R. L is an epimorphism but not a monomorphism, while R is a monomorphism but not an epimorphism. We see that L ı R D I (so R is a right inverse for L and L is a left inverse for R) but R ı L ¤ I (and neither L nor R is invertible). R/. Then D W V ! V and Ia W V ! V are linear transformations that are not invertible, but D ı Ia is the identity.

4 Subspaces and direct sum decompositions We now generalize the notion of spanning sets, linearly independent sets, and bases. We introduce the notions of V being a sum of subspaces W1 ; : : : ; Wk , of the subspaces W1 ; : : : ; Wk being independent, and of V being the direct sum of the subspaces W1 ; : : : ; Wk . In the special case where each W1 ; : : : ; Wk consists of the multiples of a single nonzero vector vi , let B D fv1 ; : : : ; vk g. 4. Subspaces and direct sum decompositions 23 V ; the subspaces W1 ; : : : ; Wk are independent if and only if B is linearly independent; and V is the direct sum of W1 ; : : : ; Wk if and only if B is a basis of V .