brinethery 3 years ago Find a basis for the space V of all skew-symmetric 3x3 matrices. What is the dimension of V?

1. brinethery

I'll be back in a sec. Brewing some coffee. Would you like some?

2. No-data

Yeah, I will in maybe two hours =P

3. No-data

Well you need to remember that a basis for any vectorial space generates it and it's vectors are linear independent.

4. No-data

And 3x3 skew-symmetric matrix is something like this:

5. No-data

$A=-A^T$

6. brinethery

:-)

7. No-data

Hmm thinking...

8. No-data

Do you have any idea to begin with?

9. No-data

Mm you need three matrices am I right?

10. No-data

Well the dimension is 3 haha

11. brinethery

I am stupid haha

12. brinethery

and I need a longer explanation b/c A=-A^T isn't gonna cut it

13. brinethery

soy estupido haha

14. No-data

uff I almost burn my brain hehe

15. brinethery

I told you linear algebra's no good!

16. No-data

Well I think that the three vectors are $\left[\begin{matrix}0 & 1 & 0\\ -1 & 0 & 0\\ 0 & 0 & 0\end{matrix}\right]$

17. No-data

Hmmm you're right

18. No-data

Is the idea of a basis clear to you?

19. No-data

Like you know why (1,0,0), (0,1,0),(0,0,1) is the basis of R^3?

20. brinethery

yes.

21. No-data

Well then you know that the basic idea behind a basis is that you can get any vector of the space by forming a linear combination.

22. No-data

like $\vec{v}=a\hat{i} + b\hat{j} + c\hat{k}$

23. No-data

|dw:1337293630890:dw|

24. No-data

well that is the idea in a graphic form.

25. No-data

If you want to obtain a basis for the 3x3 skew symmet

26. No-data

ric matrices you need to know that is the general form of matrix of that kind.

27. brinethery

okay, I'm following what you're saying

28. No-data

the definition says that the elements of an skew symmetric matrix are such as: $a_{ij}=-a_{ji}$

29. No-data

It seems that that doesn't tell us a lot about those matrices, but you can figure out what you have on the diagonal.

30. No-data

if i = j, then $a_{ii}=-a_{ji}$

31. No-data

which is true if the diagonal is made up zeros.

32. No-data

and I think that's all. An 3x3 skew symmetrical matrix has this form:

33. brinethery

Here's something... http://www.euclideanspace.com/maths/algebra/matrix/functions/skew/index.htm

34. No-data

$\left[\begin{matrix}0 & a_{12} & a_{13} \\ -a_{12} & 0 & a_{23}\\ -a_{13} & -a_{23} & a_{33} \end{matrix}\right]$

35. brinethery

Looks like it's all zeros down the diagonal.

36. No-data

shame on my haha

37. No-data

yeah a_33 = 0

38. brinethery

you're a bad math teacher haha! :-p

39. No-data

yeah! I'm terrible =(

40. No-data

but it's fun!

41. No-data

well now you can split that matrix on three parts:

42. brinethery

Yep this part is where I need explaining...

43. No-data

\left[\begin{matrix}0 & a_{12} & 0 & \\ -a_{12} & 0 & 0\\ 0&0&0\end{matrix}\right]

44. No-data

$\left[\begin{matrix}0 & 0 & a_{13} & \\ 0 & 0 & 0\\ -a_{13}&0&0\end{matrix}\right]$

45. No-data

$\left[\begin{matrix}0 & 0 & 0 & \\ 0 & 0 & a_{23}\\ 0&-a_{23}&0\end{matrix}\right]$

46. No-data

Is it clear why I did that?

47. brinethery

Oh I see. So each part of the basis will be a 3x3 matrix?

48. brinethery

And there will be (3) 3x3 matrices

49. No-data

yeah! it has to bee a 3x3 matrix

50. brinethery

I asked on cramster and some (puta) was really insulting to me AND she gave me the wrong answer.

51. No-data

what is cramster? and puta?

52. No-data

we are not finished yet.

53. No-data

you can do this:

54. brinethery

I thought it meant b)it(ch but I guess it's mexican slang for prostitute or whore

55. brinethery

but you get the idea.

56. No-data

$\left[\begin{matrix}0& 1 & 0 \\ -1 & 0 &0\\ 0&0&0\end{matrix}\right]$

57. brinethery
58. No-data

ahh I thought I did haha.

59. No-data

$\left[\begin{matrix}0& 0 & 1 \\ 0 & 0 &0\\ -1&0&0\end{matrix}\right]$

60. No-data

and \left[\begin{matrix}0& 0 & 0 \\ 0 & 0 &1\\ 0&-1&0\end{matrix}\right]

61. No-data

You multiply by the scalars a,b,c each of those matrices and form a linear combination.

62. No-data

and the result is a skew symmetrical matrix

63. No-data

I'm sorry If my explanation was awful, and my english worst =P

64. brinethery

So the dimension of all skew-symmetric matrices is 2?

65. brinethery

or that's only the dimension of the basis?

66. No-data

the dimesion is the number of vectors of the basis. so it's 3.

67. brinethery

hahaha I am so stupid!

68. brinethery

WOW. I can't believe I asked that.

69. No-data

if you have an 4x4 skew symmetrical matrix the dimension of the basis is 4.

70. brinethery

:-) yeah yeah yeah

71. No-data

Don't worry, don't judge yourself so severely.

72. No-data

Do you have any doubt regarding this brinethery?

73. brinethery

I have another question for you! No I don't have any doubt. You did a great job explaining.

74. No-data

Ok

75. No-data

Linear Algebra?