Download Abstract linear algebra by Morton L. Curtis, Paul Place PDF

By Morton L. Curtis, Paul Place

ISBN-10: 0387972633

ISBN-13: 9780387972633

Starting from scratch and constructing the traditional subject matters of Linear Algebra, this e-book is meant as a textual content for a primary path at the topic. The target to which this paintings leads is the theory of Hurwitz - that the single normed algebras over the genuine numbers are the true numbers, the complicated numbers, the quaternions, and the octonions. specific in providing this fabric at an easy point, the publication stresses the whole logical improvement of the topic and may offer a bavuable reference for mathematicians generally.

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Example text

S is called an ordinal if (S, I) is a chain. If a is an ordinal then a+ = a u { a } is also clearly an ordinal, called the successor of a. In particular 0 = 0, = 0+, 2 = i+, . . are all ordinals. O n the other hand, there are ordinals which are not successors, the first of which is {E: n E N}; these are called limit ordinals. A chain is well-ordered if every nonempty subset has a minimal element. Every ordinal a is well-ordered under < as defined above; indeed if @ # S c a then any s E S with s n S = 0 is minimal in S.

Proof: (a) as observed above. ( G )For any a1,a2in A we have fa, I f ( a l v a z ) and fa, I f ( a l v a,) so fa, v fa, I f(al v a,). Applying this argument to f - ' yields a, v a, = f-'fa, v f-lfu, I f-'(fa, v fa,) I f-'f(a, v a,) = a, v a, so equality holds at each stage. The proof for A is analogous. D. We can apply this result to the lattice 9 ( M ) . 8: Iff: M N i s onto then there is a lattice isomorphismfrom 9 ( N ) to {submodules of M containing kerf}, given by N ' -+f-'N'. (The inverse correspondence is given by M ' -+ fM').

A) Conversely suppose {eij: 1 Ii, j In} is a given set of matrix units of T. Let R = eulaelu:aE T}. R is a subring because it is closed under subtraction, and 1 = euu= x e u l l e l uE R and ( ~ e u l a e l u ) ( ~ e , l b e=u~l )e u l ( a e l l b ) eEl uR. It remains to define the euiae. = isomorphism cp: T + Mn(R). Clearly cp is an additive group homomorphism. e.. ae. e.. ae.. n u=l U1 JU lJ u= 1 U1 UI JJ I1 JJ9 xi, rijeij I;, eiiaejj (Ceii)a( ejj) a, implying is 1 1; likewise, 1rijeij) (rij)so is onto.

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