Expert: Scott A Wilson - 8/5/2009

Question
QUESTION: 1. The sum of u and v , denoted by u + v , is in V .
2. u + v = v + u .
3. (u + v) + w = u + (v + w) .
4. There is a zero vector 0 in V such that u + 0 = u .
5. For each u in V , there is a vector -u in V such that u + (-u) = 0 .
6. The scalar multiple of u by c , denoted by cu , is in V .
7. c(u + v) = cu + cv .
8. (c + d )u = cu + du .
9. c(du) = (cd)u .
10. 1u = u .

I need to redefine a vector space that only four axioms are required. Consider that the set of scalars given above could be generalized by letting the scalars be elements of a field. First five axioms all involve addition and hence constitute a special algebraic object. Lastely the closure item in # 6 can be part of the definition of scalar multiplication. How can I combine them into 4 axioms.

I have come up with this and have combined the 10 axioms into 4
A vector space over a field F is an additive abelian group (V, +)   and a map  F x  ,   called scalar multiplication, (where the image of the pair (c, v) for c in F and v in V is denoted simply cv)  such that, for  all  v, w in V and  c, d in F.
1. c(v + w)  = cv + cw
2. (c + d)v =  cv + dv
3. (cd)v = c(dv)
4. 1v = v (where 1 is the multiplicative identity of F)

Do you know why the other axioms are not needed and what algebraic properties or laws were used.

ANSWER: I'm not entirly sure of every answer, but I do have somewhat of a grasp of how to do them.

Looking at the 4 vectors space rules, I don't see how they can apply.

1. The sum of u and v , denoted by u + v , is in V .
What can be said is if u, v are in V, then u+v is also in V.
It can also be said that if v, u are in V, then v+u is in V.
Both of these look to come off the first law with c=1.

2. u + v = v + u.
It seems like I just answered that in 1.

3. (u + v) + w = u + (v + w) .
Use rule 1 with c=1 again.  Use rule 4 to say 1w = w.
(u+v) + w = 1(u+v) + 1w = 1u + 1v + 1w = 1u +1(v+w) = u + (v+w)

4. There is a zero vector 0 in V such that u + 0 = u .
Suppose you take rule 2 with c = -d.  Since c - d = 0, this is the zero vect that is found.

5. For each u in V , there is a vector -u in V such that u + (-u) = 0 .
See answer to 4.

6. The scalar multiple of u by c , denoted by cu , is in V .  
Let w be the zero vector.  From 1 I can say that c(v+0) = cv + 0v = cv.

7. c(u + v) = cu + cv .
Repeat of Rule 1

8. (c + d )u = cu + du .
Repeat of Rule 2

9. c(du) = (cd)u .
Repeat of Rule 3

10. 1u = u .
Repeat of Rule 4


---------- FOLLOW-UP ----------

QUESTION: I have come up with this and have combined the 10 axioms into 4
A vector space over a field F is an additive abelian group (V, +)   and a map  F x  ,   called scalar multiplication, (where the image of the pair (c, v) for c in F and v in V is denoted simply cv)  such that, for  all  v, w in V and  c, d in F.
1. c(v + w)  = cv + cw
2. (c + d)v =  cv + dv
3. (cd)v = c(dv)
4. 1v = v (where 1 is the multiplicative identity of F)

Do you know why the other axioms are not needed and what algebraic properties or laws were used.

I had to use definitions from abstract algebra, so that is how I shortened the axioms from 10 to 4 axioms, but I am stuck on the algebraic properties or laws for the 4 axioms. Why are the other 6 not needed.

Answer
The reason they are needed is the same reason that multiplication is needed.
After all, when you multiply something by 5, you could just as well add it together 5 times.
Multiplication is a quicker way of doing it, but it is not needed.

When you want to find the 4th angle of a quadralateral, you could go through the proof to show that the sum of the angles of a triangle equals 180° and then show that a quadralateral is the sum of two triangles, and then show that the sum of angles of a quadralateral is 360°.  Once this has been done, you could find the angle.  That is not done, though.  People just reference the proof that the sum of angles in a n-gon is (n-2)180.

In physics, someone could always derive the speed of fall every time an experiment is done.  However, most just use the fact that is is 32 ft/s² or 9.8 m/s².

In calculus, someone could prove that ∫sinx dx = cosx + C every time they did it, but all people who have had calculus know it is true and don't care to see the proof again.  There are a few people that even remember how it goes, but who cares?  Calculus says its true.

Similarly, in using the axioms above in higher dimensional math, the axiom is used instead of going back over the proof every time it comes up.

In algebra, the theorem that says that x = ±√)(-√b² - 4ac)/(2a)) is used quite a bit,
but the derivation of that formula is not remembered every time it is used.

I have dealt a lot with numerical analysis and not so much with abstract algebra,
but I do know how it goes.  In numerical analysis, people use theorem an awful lot without reiterating the proof every time.  For example, Newton's Method states that
x sub n+1 = x sub n - f(x sub n)/f'(x sub n).  Now and then I review the proof of how to get there in my head, but quickly it comes back to my memory and I don't write down the proof.

If I wanted to, I could write several more areas in which theorems that have been proved are used to advance to other theorems, but I think that what I said was enough.  If you are really interested in why the theorems are used, go to the internet and type in the name of the theorem.
If it doesn't have a name, that is probably because it is not used that much.

I typed "zero vector 0 in V such that u + 0 = u " in the search box (now that is really long) and found the reference http://math.gmu.edu/~dwalnut/teach/Math203/Summer08/sec4_1.pdf

I typed "c(u + v) = cu + cv " in the search box and got 8 references where it occurs.

Basically, theorems that are known to be true are used
to make future work in the area much easier.