(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let F be any field, and fix a є F. Equip the set V = F^{2}with two operations as

follows. Define addition by

(x, y)‡(x', y') := (x + x', y + y' − a), for all x, x', y, y' є F,

and define the scalar multiplication by scalars by

c * (x, y) := (cx, cy − ac + a), for all x, y, c є F.

(i) Prove that F^{2}, with these two operations satisfies the two existence axioms for a vector space over F.

3. The attempt at a solution

i) The existance of a zero vector: there exists a zero vector such that 0 + v = v

So I let (x', y') be the zero vector = (0, 0), and got

(x,y) ‡ (0, 0) := (x + 0, y + 0 - a)

= (x, y-a)

At this point, I'm not sure how to deal with the a, or maybe my whole process is wrong! ANy help is appreciated. I'm having the same problem for the other axiom - existance of a negative.

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# Homework Help: Linear Algebra-Fields and axioms

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