## anonymous 4 years ago Find a basis for the space V of all skew-symmetric 3x3 matrices. What is the dimension of V?

1. anonymous

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

2. anonymous

Yeah, I will in maybe two hours =P

3. anonymous

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

4. anonymous

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

5. anonymous

$A=-A^T$

6. anonymous

:-)

7. anonymous

Hmm thinking...

8. anonymous

Do you have any idea to begin with?

9. anonymous

Mm you need three matrices am I right?

10. anonymous

Well the dimension is 3 haha

11. anonymous

I am stupid haha

12. anonymous

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

13. anonymous

soy estupido haha

14. anonymous

uff I almost burn my brain hehe

15. anonymous

I told you linear algebra's no good!

16. anonymous

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

17. anonymous

Hmmm you're right

18. anonymous

Is the idea of a basis clear to you?

19. anonymous

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

20. anonymous

yes.

21. anonymous

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. anonymous

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

23. anonymous

|dw:1337293630890:dw|

24. anonymous

well that is the idea in a graphic form.

25. anonymous

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

26. anonymous

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

27. anonymous

okay, I'm following what you're saying

28. anonymous

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

29. anonymous

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. anonymous

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

31. anonymous

which is true if the diagonal is made up zeros.

32. anonymous

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

33. anonymous

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

34. anonymous

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

35. anonymous

Looks like it's all zeros down the diagonal.

36. anonymous

shame on my haha

37. anonymous

yeah a_33 = 0

38. anonymous

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

39. anonymous

yeah! I'm terrible =(

40. anonymous

but it's fun!

41. anonymous

well now you can split that matrix on three parts:

42. anonymous

Yep this part is where I need explaining...

43. anonymous

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

44. anonymous

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

45. anonymous

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

46. anonymous

Is it clear why I did that?

47. anonymous

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

48. anonymous

And there will be (3) 3x3 matrices

49. anonymous

yeah! it has to bee a 3x3 matrix

50. anonymous

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

51. anonymous

what is cramster? and puta?

52. anonymous

we are not finished yet.

53. anonymous

you can do this:

54. anonymous

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

55. anonymous

but you get the idea.

56. anonymous

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

57. anonymous
58. anonymous

ahh I thought I did haha.

59. anonymous

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

60. anonymous

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

61. anonymous

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

62. anonymous

and the result is a skew symmetrical matrix

63. anonymous

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

64. anonymous

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

65. anonymous

or that's only the dimension of the basis?

66. anonymous

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

67. anonymous

hahaha I am so stupid!

68. anonymous

WOW. I can't believe I asked that.

69. anonymous

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

70. anonymous

:-) yeah yeah yeah

71. anonymous

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

72. anonymous

Do you have any doubt regarding this brinethery?

73. anonymous

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

74. anonymous

Ok

75. anonymous

Linear Algebra?