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

- anonymous

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

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- anonymous

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

- anonymous

Yeah, I will in maybe two hours =P

- anonymous

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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- anonymous

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

- anonymous

\[A=-A^T\]

- anonymous

:-)

- anonymous

Hmm thinking...

- anonymous

Do you have any idea to begin with?

- anonymous

Mm you need three matrices am I right?

- anonymous

Well the dimension is 3 haha

- anonymous

I am stupid haha

- anonymous

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

- anonymous

soy estupido haha

- anonymous

uff
I almost burn my brain hehe

- anonymous

I told you linear algebra's no good!

- anonymous

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

- anonymous

Hmmm you're right

- anonymous

Is the idea of a basis clear to you?

- anonymous

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

- anonymous

yes.

- 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.

- anonymous

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

- anonymous

|dw:1337293630890:dw|

- anonymous

well that is the idea in a graphic form.

- anonymous

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

- anonymous

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

- anonymous

okay, I'm following what you're saying

- anonymous

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

- 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.

- anonymous

if i = j, then \[a_{ii}=-a_{ji}\]

- anonymous

which is true if the diagonal is made up zeros.

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

Looks like it's all zeros down the diagonal.

- anonymous

shame on my haha

- anonymous

yeah a_33 = 0

- anonymous

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

- anonymous

yeah! I'm terrible =(

- anonymous

but it's fun!

- anonymous

well now you can split that matrix on three parts:

- anonymous

Yep this part is where I need explaining...

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

Is it clear why I did that?

- anonymous

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

- anonymous

And there will be (3) 3x3 matrices

- anonymous

yeah! it has to bee a 3x3 matrix

- anonymous

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

- anonymous

what is cramster? and puta?

- anonymous

we are not finished yet.

- anonymous

you can do this:

- anonymous

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

- anonymous

but you get the idea.

- anonymous

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

- anonymous

http://www.chegg.com/homework-help/questions-and-answers/basis-space-v-skew-symmetric-3x3-matrices-dimension-v-q2552939

- anonymous

ahh I thought I did haha.

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

and the result is a skew symmetrical matrix

- anonymous

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

- anonymous

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

- anonymous

or that's only the dimension of the basis?

- anonymous

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

- anonymous

hahaha I am so stupid!

- anonymous

WOW. I can't believe I asked that.

- anonymous

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

- anonymous

:-) yeah yeah yeah

- anonymous

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

- anonymous

Do you have any doubt regarding this brinethery?

- anonymous

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

- anonymous

Ok

- anonymous

Linear Algebra?

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