anonymous
  • anonymous
Find a basis for the space V of all skew-symmetric 3x3 matrices. What is the dimension of V?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
I'll be back in a sec. Brewing some coffee. Would you like some?
anonymous
  • anonymous
Yeah, I will in maybe two hours =P
anonymous
  • 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
  • anonymous
And 3x3 skew-symmetric matrix is something like this:
anonymous
  • anonymous
\[A=-A^T\]
anonymous
  • anonymous
:-)
anonymous
  • anonymous
Hmm thinking...
anonymous
  • anonymous
Do you have any idea to begin with?
anonymous
  • anonymous
Mm you need three matrices am I right?
anonymous
  • anonymous
Well the dimension is 3 haha
anonymous
  • anonymous
I am stupid haha
anonymous
  • anonymous
and I need a longer explanation b/c A=-A^T isn't gonna cut it
anonymous
  • anonymous
soy estupido haha
anonymous
  • anonymous
uff I almost burn my brain hehe
anonymous
  • anonymous
I told you linear algebra's no good!
anonymous
  • 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
  • anonymous
Hmmm you're right
anonymous
  • anonymous
Is the idea of a basis clear to you?
anonymous
  • anonymous
Like you know why (1,0,0), (0,1,0),(0,0,1) is the basis of R^3?
anonymous
  • anonymous
yes.
anonymous
  • 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
  • anonymous
like \[\vec{v}=a\hat{i} + b\hat{j} + c\hat{k}\]
anonymous
  • anonymous
|dw:1337293630890:dw|
anonymous
  • anonymous
well that is the idea in a graphic form.
anonymous
  • anonymous
If you want to obtain a basis for the 3x3 skew symmet
anonymous
  • anonymous
ric matrices you need to know that is the general form of matrix of that kind.
anonymous
  • anonymous
okay, I'm following what you're saying
anonymous
  • anonymous
the definition says that the elements of an skew symmetric matrix are such as: \[a_{ij}=-a_{ji}\]
anonymous
  • 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
  • anonymous
if i = j, then \[a_{ii}=-a_{ji}\]
anonymous
  • anonymous
which is true if the diagonal is made up zeros.
anonymous
  • anonymous
and I think that's all. An 3x3 skew symmetrical matrix has this form:
anonymous
  • anonymous
Here's something... http://www.euclideanspace.com/maths/algebra/matrix/functions/skew/index.htm
anonymous
  • anonymous
\[\left[\begin{matrix}0 & a_{12} & a_{13} \\ -a_{12} & 0 & a_{23}\\ -a_{13} & -a_{23} & a_{33} \end{matrix}\right]\]
anonymous
  • anonymous
Looks like it's all zeros down the diagonal.
anonymous
  • anonymous
shame on my haha
anonymous
  • anonymous
yeah a_33 = 0
anonymous
  • anonymous
you're a bad math teacher haha! :-p
anonymous
  • anonymous
yeah! I'm terrible =(
anonymous
  • anonymous
but it's fun!
anonymous
  • anonymous
well now you can split that matrix on three parts:
anonymous
  • anonymous
Yep this part is where I need explaining...
anonymous
  • anonymous
\left[\begin{matrix}0 & a_{12} & 0 & \\ -a_{12} & 0 & 0\\ 0&0&0\end{matrix}\right]
anonymous
  • anonymous
\[\left[\begin{matrix}0 & 0 & a_{13} & \\ 0 & 0 & 0\\ -a_{13}&0&0\end{matrix}\right]\]
anonymous
  • anonymous
\[\left[\begin{matrix}0 & 0 & 0 & \\ 0 & 0 & a_{23}\\ 0&-a_{23}&0\end{matrix}\right]\]
anonymous
  • anonymous
Is it clear why I did that?
anonymous
  • anonymous
Oh I see. So each part of the basis will be a 3x3 matrix?
anonymous
  • anonymous
And there will be (3) 3x3 matrices
anonymous
  • anonymous
yeah! it has to bee a 3x3 matrix
anonymous
  • anonymous
I asked on cramster and some (puta) was really insulting to me AND she gave me the wrong answer.
anonymous
  • anonymous
what is cramster? and puta?
anonymous
  • anonymous
we are not finished yet.
anonymous
  • anonymous
you can do this:
anonymous
  • anonymous
I thought it meant b)it(ch but I guess it's mexican slang for prostitute or whore
anonymous
  • anonymous
but you get the idea.
anonymous
  • anonymous
\[\left[\begin{matrix}0& 1 & 0 \\ -1 & 0 &0\\ 0&0&0\end{matrix}\right]\]
anonymous
  • anonymous
http://www.chegg.com/homework-help/questions-and-answers/basis-space-v-skew-symmetric-3x3-matrices-dimension-v-q2552939
anonymous
  • anonymous
ahh I thought I did haha.
anonymous
  • anonymous
\[\left[\begin{matrix}0& 0 & 1 \\ 0 & 0 &0\\ -1&0&0\end{matrix}\right]\]
anonymous
  • anonymous
and \left[\begin{matrix}0& 0 & 0 \\ 0 & 0 &1\\ 0&-1&0\end{matrix}\right]
anonymous
  • anonymous
You multiply by the scalars a,b,c each of those matrices and form a linear combination.
anonymous
  • anonymous
and the result is a skew symmetrical matrix
anonymous
  • anonymous
I'm sorry If my explanation was awful, and my english worst =P
anonymous
  • anonymous
So the dimension of all skew-symmetric matrices is 2?
anonymous
  • anonymous
or that's only the dimension of the basis?
anonymous
  • anonymous
the dimesion is the number of vectors of the basis. so it's 3.
anonymous
  • anonymous
hahaha I am so stupid!
anonymous
  • anonymous
WOW. I can't believe I asked that.
anonymous
  • anonymous
if you have an 4x4 skew symmetrical matrix the dimension of the basis is 4.
anonymous
  • anonymous
:-) yeah yeah yeah
anonymous
  • anonymous
Don't worry, don't judge yourself so severely.
anonymous
  • anonymous
Do you have any doubt regarding this brinethery?
anonymous
  • anonymous
I have another question for you! No I don't have any doubt. You did a great job explaining.
anonymous
  • anonymous
Ok
anonymous
  • anonymous
Linear Algebra?

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