## brinethery 3 years ago Linear Algebra: Determine whether this is a vector space. Either show that the necessary properties are satisfied, or give an example that at least one of them is not. -The set of all upper-triangular m x n matrices

1. brinethery

Ah, I just don't get this stuff!

2. Pippa

hehe wait a sec i am just checking :)

3. KingGeorge

I'm pretty sure this is a vector space since it satisfies all the axioms need for a vector space.

4. Pippa

it is a vector space that is forsure

5. Pippa

you just have to go thru all the axioms which is a pain :)

6. Pippa

umm i wonder if you can assign values to the matrix to show that all axioms are true or you just have to leave them as variables

7. Pippa

umm if you show using real values it is easier

8. Pippa

but i am not sure what ur prof wld want

9. brinethery

It would probably have to be more formal mathematical language... he docked me when I was doing a proof with a 3x3 matrix one time.

10. KingGeorge

All you would really need to show is that the set of upper triangular matrices is closed under addition which is obvious. The other axioms are properties of matrices, or follow directly from closure of addition.

11. Pippa

well i doubt her prof wld accepy that :)

12. Pippa

anyways if its closed under addition not neccesairy will it be cllosed under multiplication

13. KingGeorge

Well, closure of addition implies that u+v is still a matrix, if u, v are upper-triangular matrices. Since addition is associative, and multiplying matrices by scalars is a property of matrices that is well-defined, all the axioms are satisfied. In a vector space, you never multiply vectors together.

14. Pippa

kk but she still has to show that all the axioms are satisfied

15. KingGeorge

Yes, but their proof are almost trivial once you've showed closure of addition.

16. Pippa

True :)

17. Pippa

umm brinethy u know how to show that all the 10 axioms r true?

18. brinethery

I don't know how to do this with a general mxn matrix

19. brinethery

showing that it's closed under scalar multiplication and vector addition, I mean.

20. Pippa

give me a sec and i will do it

21. KingGeorge

Scalar multiplication should already be given to you as a property of matrices. As for vector addition, choose two arbitrary m x n upper triangular matrices, and add them together. It should become pretty obvious that it's closed after that.

22. brinethery

Pippa, thank you very much for taking the time to do this for me. I can't tell you how much I appreciate it.

23. Pippa

24. Pippa

well i just proved that they are closed under addition and scaler multiplication. Do u see how?

25. KingGeorge

Aren't those all lower triangular matrices? not upper?

26. Pippa

oh Sh* but it wld be the same thing

27. Pippa

its a good thing u r around king

28. KingGeorge

no problem at all.

29. Pippa

umm there are still 8 more axioms that need to be proved

30. Pippa

some of them take much longer to prive than others

31. Pippa

These are the two main ones

32. Pippa

R u still there?

33. brinethery

34. brinethery

I'm opening your file right now

35. Pippa

lol that is ok:)

36. brinethery

So as long as it's closed under vector addition and scalar multiplication, the set is a vector space?

37. Pippa

umm well u kind of have to prove the other axioms as well which is harder

38. Pippa

and btw by accident i proved using lower traingles in stead of upper

39. brinethery

Yes, I am aware of that

40. KingGeorge

While you have to show a little more, the only other part that doesn't follow directly from those facts is that you need to find an inverse matrix (hint: multiply by -1 to get the inverse)

41. Pippa

y wld u need to do that?

42. Pippa

I am just wondering cuz i dont remember ever doing that?

43. Pippa

idk I just went thru every axiom

44. brinethery

Sorry guys, I am even more lost. I guess you have to go through all the axioms but I'm not getting a clear answer. I'll ask the instructor tomorrow, but he'll probably confuse me even more lol

45. Pippa

LOL yes well you just went thru two axioms you just proved that it is closed under addition and that it is closed under scaler mutiplication

46. Pippa

Like in ur textbook there shld be another 8 axioms that need to be proved liek one of them is u+v = v+u and (u+v)+w=u+(w+v) and many others

47. Pippa

but it will take me hours to write them out unfortunately i dont have a tablet so like maybe ask ur prof to show u how

48. Pippa

k gonna go study i hope i was a lil bit helpful :)

49. brinethery

I will. And I really appreciate your help, thank you for taking the time to explain this stuff to me.

50. Pippa

U r welcome its fun to teach others cuz that means I am comfortable enough in this area to explain it to others :) Gluck