By Robert J. Valenza

Based on lectures given at Claremont McKenna collage, this article constitutes a considerable, summary advent to linear algebra. The presentation emphasizes the structural parts over the computational - for instance by means of connecting matrices to linear variations from the outset - and prepares the coed for additional learn of summary arithmetic. Uniquely between algebra texts at this point, it introduces team idea early within the dialogue, to illustrate of the rigorous improvement of casual axiomatic systems.

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**Example text**

Then by the definition of an identity for the group operation, we have e = ee' = e', which proves uniqueness. Note that this argument works in any magma. PROOF. 22 2. Groups and Group Homomorphisms (ii) Suppose that both t and t' are inverses for SE G. Then st = e = st', and by left cancellation we have t=t'. (iii) Suppose that st=e. Multiply both sides by s-I to find that which via associativity shows at once that S-I = t. (iv) We have already observed that if t is an inverse for s, then s is an inverse for t.

In particular, R is itself a real vector space; C is itself a complex vector space. (2) Let k be any field and let k 2 denote the set of all ordered pairs (X 1,X2) of elements of k. Then k2 is a vector space over k with respect to the operations (XI'X2) + (Yl,y2) = (x 1+Yl'x2+Y2) V'XI'X2,yl,y2 Ek a(x1,x2) = ( axl'ax2) V'a,x 1,x2Ek We know that k 2 is an additive group by Exercise 5 of the previous chapter, and the other axioms are inherited directly from the field axioms for k. Note in particular that the additive group identity is (0,0); the additive inverse of (xl'x2) is (-xl'-x2).

In this case we say a and b are zero divisors. 22. Show that a field can never have zero divisors; that is, if ab =0, then either a or b is itself equal to o. 23. Write out both Cayley tables (one for addition, and one for multiplication) for F s. ) 24. Write out both Cayley tables for Z6' the ring of integers modulo 6. Show that this ring is not a field. 25. More generally show that Zn is not a field whenever n is not a prime integer. ) 26. A commutative ring without zero divisors is called an integral domain.