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

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brinethery Group TitleBest ResponseYou've already chosen the best response.0
I'll be back in a sec. Brewing some coffee. Would you like some?
 2 years ago

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Yeah, I will in maybe two hours =P
 2 years ago

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

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And 3x3 skewsymmetric matrix is something like this:
 2 years ago

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Hmm thinking...
 2 years ago

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Do you have any idea to begin with?
 2 years ago

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Mm you need three matrices am I right?
 2 years ago

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Well the dimension is 3 haha
 2 years ago

brinethery Group TitleBest ResponseYou've already chosen the best response.0
I am stupid haha
 2 years ago

brinethery Group TitleBest ResponseYou've already chosen the best response.0
and I need a longer explanation b/c A=A^T isn't gonna cut it
 2 years ago

brinethery Group TitleBest ResponseYou've already chosen the best response.0
soy estupido haha
 2 years ago

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uff I almost burn my brain hehe
 2 years ago

brinethery Group TitleBest ResponseYou've already chosen the best response.0
I told you linear algebra's no good!
 2 years ago

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

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Hmmm you're right
 2 years ago

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Is the idea of a basis clear to you?
 2 years ago

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

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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.
 2 years ago

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

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dw:1337293630890:dw
 2 years ago

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well that is the idea in a graphic form.
 2 years ago

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

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

brinethery Group TitleBest ResponseYou've already chosen the best response.0
okay, I'm following what you're saying
 2 years ago

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

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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.
 2 years ago

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if i = j, then \[a_{ii}=a_{ji}\]
 2 years ago

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which is true if the diagonal is made up zeros.
 2 years ago

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

brinethery Group TitleBest ResponseYou've already chosen the best response.0
Here's something... http://www.euclideanspace.com/maths/algebra/matrix/functions/skew/index.htm
 2 years ago

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

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Looks like it's all zeros down the diagonal.
 2 years ago

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shame on my haha
 2 years ago

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yeah a_33 = 0
 2 years ago

brinethery Group TitleBest ResponseYou've already chosen the best response.0
you're a bad math teacher haha! :p
 2 years ago

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yeah! I'm terrible =(
 2 years ago

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but it's fun!
 2 years ago

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well now you can split that matrix on three parts:
 2 years ago

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Yep this part is where I need explaining...
 2 years ago

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\left[\begin{matrix}0 & a_{12} & 0 & \\ a_{12} & 0 & 0\\ 0&0&0\end{matrix}\right]
 2 years ago

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\[\left[\begin{matrix}0 & 0 & a_{13} & \\ 0 & 0 & 0\\ a_{13}&0&0\end{matrix}\right]\]
 2 years ago

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\[\left[\begin{matrix}0 & 0 & 0 & \\ 0 & 0 & a_{23}\\ 0&a_{23}&0\end{matrix}\right]\]
 2 years ago

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Is it clear why I did that?
 2 years ago

brinethery Group TitleBest ResponseYou've already chosen the best response.0
Oh I see. So each part of the basis will be a 3x3 matrix?
 2 years ago

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And there will be (3) 3x3 matrices
 2 years ago

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yeah! it has to bee a 3x3 matrix
 2 years ago

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

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what is cramster? and puta?
 2 years ago

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we are not finished yet.
 2 years ago

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you can do this:
 2 years ago

brinethery Group TitleBest ResponseYou've already chosen the best response.0
I thought it meant b)it(ch but I guess it's mexican slang for prostitute or whore
 2 years ago

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but you get the idea.
 2 years ago

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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.
 2 years ago

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\[\left[\begin{matrix}0& 0 & 1 \\ 0 & 0 &0\\ 1&0&0\end{matrix}\right]\]
 2 years ago

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and \left[\begin{matrix}0& 0 & 0 \\ 0 & 0 &1\\ 0&1&0\end{matrix}\right]
 2 years ago

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

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and the result is a skew symmetrical matrix
 2 years ago

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

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So the dimension of all skewsymmetric matrices is 2?
 2 years ago

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or that's only the dimension of the basis?
 2 years ago

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

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hahaha I am so stupid!
 2 years ago

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WOW. I can't believe I asked that.
 2 years ago

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

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:) yeah yeah yeah
 2 years ago

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Don't worry, don't judge yourself so severely.
 2 years ago

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Do you have any doubt regarding this brinethery?
 2 years ago

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

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Linear Algebra?
 2 years ago
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