## anilorap 3 years ago LINEAR ALGEBRA. verifying (k+m)(u)=ku+mu... where k,m are any integers and u is an element of V={(x,y,z) | z=3}. addition defined by (x1,y1,z1)+(x2,y2,z2) = (x1+x2,y1+y2,3) scalar multiplication defined by s(x,y,z)=(sx,sy,3)

1. imranmeah91

I am learning linear algebra as well, can you tell me what topic this comes from ?

2. anilorap

determining is v is a vector space... so we have to verify the ten axioms

3. imranmeah91

so we are finding what order space this is?

4. anilorap

mm.. what u mean with order space?

5. imranmeah91

line plane is 2nd order space, line is 1st order space

6. anilorap

ohhh,,, well my example is already i 3 space

7. anilorap

if u already covered this section.. we defined vector space as a set of vectors on which are defined by two operations: 1) vector addition 2) vector multiplication and must satisfy 10 axioms

8. anilorap

2)scalar multiplication***

9. imranmeah91

I know about first two , but never heard of this '10 axioms' stuff

10. imranmeah91

are you learning it independently or with class?

11. anilorap

class

12. anilorap

1. u + v 2 V (V is closed under addition) 2. u + v = v + u (commutative property) 3. (u + v) + w = u + (v + w) (associative property) 4. There exists a vector in V , called the zero vector and denoted 0 such that u + 0 = u (additive identity) 5. For every vector u in V , there exists a vector 􀀀u also in V such that u + (􀀀u) = 0 (additive inverse) 6. cu 2 V (V is closed under scalar multiplication) 7. c (u + v) = cu + cv 8. (c + d) u = cu + du 9. c (du) = (cd) u 10. 1u = u

13. anilorap

so in this case im trying to verify number 8

14. imranmeah91

distributive property

15. anilorap

kind of

16. imranmeah91

do you watch MIT lectures

17. anilorap

mmm nop

18. anilorap

are u learning independenly

19. imranmeah91

yes

20. anilorap

ohh

21. imranmeah91

I tend to do better in class , if I already learned it beforehand

22. imranmeah91

have you done differential equation yet?

23. anilorap

yes in calculus..

24. anilorap

im learning it now

25. imranmeah91

let me know if you need any help with differential equation; I am really good at it

26. anilorap

:).. cool... i like derivative for some reasons... where im having a little trouble is in double and triple integrals... are u also learning that independnly

27. imranmeah91

no, I just finished the class this semseter

28. anilorap

so u r good with double and triples integrals too?

29. mathmate

Given u(x,y,z), (x1,y1,z1)+(x2,y2,z2) = (x1+x2,y1+y2,3) s(x,y,z)=(sx,sy,3) Assuming vector space is defined over field F such as R. (k+m)(u) =(k+m)(x,y,3) [definition of u] =((k+m)x,(k+m)y,3) [definition of scalar multiplication] =(kx+mx, ky+my,3) [distributivity defined over F] =(kx,ky,3)+(mx,my,3) [definition of addition] =k(x,y,3)+m(x,y,3) [definition of scalar multiplication] =ku+mu [definition of u] QED