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brinethery

  • 2 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

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  1. brinethery
    • 2 years ago
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    Ah, I just don't get this stuff!

  2. Pippa
    • 2 years ago
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    hehe wait a sec i am just checking :)

  3. KingGeorge
    • 2 years ago
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    I'm pretty sure this is a vector space since it satisfies all the axioms need for a vector space.

  4. Pippa
    • 2 years ago
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    it is a vector space that is forsure

  5. Pippa
    • 2 years ago
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    you just have to go thru all the axioms which is a pain :)

  6. Pippa
    • 2 years ago
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    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
    • 2 years ago
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    umm if you show using real values it is easier

  8. Pippa
    • 2 years ago
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    but i am not sure what ur prof wld want

  9. brinethery
    • 2 years ago
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    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
    • 2 years ago
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    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
    • 2 years ago
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    well i doubt her prof wld accepy that :)

  12. Pippa
    • 2 years ago
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    anyways if its closed under addition not neccesairy will it be cllosed under multiplication

  13. KingGeorge
    • 2 years ago
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    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
    • 2 years ago
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    kk but she still has to show that all the axioms are satisfied

  15. KingGeorge
    • 2 years ago
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    Yes, but their proof are almost trivial once you've showed closure of addition.

  16. Pippa
    • 2 years ago
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    True :)

  17. Pippa
    • 2 years ago
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    umm brinethy u know how to show that all the 10 axioms r true?

  18. brinethery
    • 2 years ago
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    I don't know how to do this with a general mxn matrix

  19. brinethery
    • 2 years ago
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    showing that it's closed under scalar multiplication and vector addition, I mean.

  20. Pippa
    • 2 years ago
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    give me a sec and i will do it

  21. KingGeorge
    • 2 years ago
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    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
    • 2 years ago
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    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
    • 2 years ago
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  24. Pippa
    • 2 years ago
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    well i just proved that they are closed under addition and scaler multiplication. Do u see how?

  25. KingGeorge
    • 2 years ago
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    Aren't those all lower triangular matrices? not upper?

  26. Pippa
    • 2 years ago
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    oh Sh* but it wld be the same thing

  27. Pippa
    • 2 years ago
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    its a good thing u r around king

  28. KingGeorge
    • 2 years ago
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    no problem at all.

  29. Pippa
    • 2 years ago
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    umm there are still 8 more axioms that need to be proved

  30. Pippa
    • 2 years ago
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    some of them take much longer to prive than others

  31. Pippa
    • 2 years ago
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    These are the two main ones

  32. Pippa
    • 2 years ago
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    R u still there?

  33. brinethery
    • 2 years ago
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    Yeah, I stepped out. My dad had to tell me something

  34. brinethery
    • 2 years ago
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    I'm opening your file right now

  35. Pippa
    • 2 years ago
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    lol that is ok:)

  36. brinethery
    • 2 years ago
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    So as long as it's closed under vector addition and scalar multiplication, the set is a vector space?

  37. Pippa
    • 2 years ago
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    umm well u kind of have to prove the other axioms as well which is harder

  38. Pippa
    • 2 years ago
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    and btw by accident i proved using lower traingles in stead of upper

  39. brinethery
    • 2 years ago
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    Yes, I am aware of that

  40. KingGeorge
    • 2 years ago
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    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
    • 2 years ago
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    y wld u need to do that?

  42. Pippa
    • 2 years ago
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    I am just wondering cuz i dont remember ever doing that?

  43. Pippa
    • 2 years ago
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    idk I just went thru every axiom

  44. brinethery
    • 2 years ago
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    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
    • 2 years ago
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    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
    • 2 years ago
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    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
    • 2 years ago
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    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
    • 2 years ago
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    k gonna go study i hope i was a lil bit helpful :)

  49. brinethery
    • 2 years ago
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    I will. And I really appreciate your help, thank you for taking the time to explain this stuff to me.

  50. Pippa
    • 2 years ago
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    U r welcome its fun to teach others cuz that means I am comfortable enough in this area to explain it to others :) Gluck

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