## brinethery Group Title Find a basis for the space V of all skew-symmetric 3x3 matrices. What is the dimension of V? 2 years ago 2 years ago

1. brinethery Group Title

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

2. No-data Group Title

Yeah, I will in maybe two hours =P

3. No-data Group Title

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

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

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$A=-A^T$

6. brinethery Group Title

:-)

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Hmm thinking...

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Do you have any idea to begin with?

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Mm you need three matrices am I right?

10. No-data Group Title

Well the dimension is 3 haha

11. brinethery Group Title

I am stupid haha

12. brinethery Group Title

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

13. brinethery Group Title

soy estupido haha

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uff I almost burn my brain hehe

15. brinethery Group Title

I told you linear algebra's no good!

16. No-data Group Title

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 Group Title

Hmmm you're right

18. No-data Group Title

Is the idea of a basis clear to you?

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

20. brinethery Group Title

yes.

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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.

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

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|dw:1337293630890:dw|

24. No-data Group Title

well that is the idea in a graphic form.

25. No-data Group Title

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

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

27. brinethery Group Title

okay, I'm following what you're saying

28. No-data Group Title

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

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

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

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which is true if the diagonal is made up zeros.

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

33. brinethery Group Title

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

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

35. brinethery Group Title

Looks like it's all zeros down the diagonal.

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shame on my haha

37. No-data Group Title

yeah a_33 = 0

38. brinethery Group Title

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

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yeah! I'm terrible =(

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but it's fun!

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well now you can split that matrix on three parts:

42. brinethery Group Title

Yep this part is where I need explaining...

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

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

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

46. No-data Group Title

Is it clear why I did that?

47. brinethery Group Title

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

48. brinethery Group Title

And there will be (3) 3x3 matrices

49. No-data Group Title

yeah! it has to bee a 3x3 matrix

50. brinethery Group Title

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

51. No-data Group Title

what is cramster? and puta?

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we are not finished yet.

53. No-data Group Title

you can do this:

54. brinethery Group Title

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

55. brinethery Group Title

but you get the idea.

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

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ahh I thought I did haha.

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

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

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

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and the result is a skew symmetrical matrix

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

64. brinethery Group Title

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

65. brinethery Group Title

or that's only the dimension of the basis?

66. No-data Group Title

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

67. brinethery Group Title

hahaha I am so stupid!

68. brinethery Group Title

WOW. I can't believe I asked that.

69. No-data Group Title

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

70. brinethery Group Title

:-) yeah yeah yeah

71. No-data Group Title

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

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Do you have any doubt regarding this brinethery?

73. brinethery Group Title

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

74. No-data Group Title

Ok

75. No-data Group Title

Linear Algebra?