1. ## Assistance with Subspaces of General Vector Spaces please

Hello

Some questions I have here
https://imgur.com/a/TRLBe

3 b) is dimension 8 right? and subspace 6 matrices with 1's on each point and zero on others except diagonals and 2 matrices with 1 and -1 on diagonals?

then 3c and 3d I have no idea haha

Plox assist

2. ## Re: Assistance with Subspaces of General Vector Spaces please

3b) not quite

$W_{3,3} = -(a+e)$

so we have

$a\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}+ b\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}+ c\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix}+ d\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\end{pmatrix}+ e\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}+ f\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}+ g\begin{pmatrix}0&0&0\\0&0&0\\1&0&0\end{pmatrix}+ h\begin{pmatrix}0&0&0\\0&0&0\\0&1&0\end{pmatrix}+ (a+e)\begin{pmatrix}0&0&0\\0&0&0\\0&0&-1\end{pmatrix} =$

$a\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}+ b\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}+ c\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix}+ d\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\end{pmatrix}+ e\begin{pmatrix}0&0&0\\0&1&0\\0&0&-1\end{pmatrix}+ f\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}+ g\begin{pmatrix}0&0&0\\0&0&0\\1&0&0\end{pmatrix}+ h\begin{pmatrix}0&0&0\\0&0&0\\0&1&0\end{pmatrix}$

and yes it's of dimension 8

3c) A little thought will show that $W = \begin{pmatrix}0 &a &b \\ -a &0 &c \\-b &-c &0 \end{pmatrix}$

you should be able to apply the method of (3b) to this to come up with the basis and dimension

$dim(V)=9,~dim(W)=3,~codim_V(W)=9-3=6$ There are also 6 conditions on $W$
$W_{1,1}=W_{2,2}=W_{3,3}=0$
$W_{2,1}=-W_{1,2},~W_{3,1}=-W_{1,3},~W_{3,2}=-W_{2,3}$