Hi

Can someone kindly help me out with this and check

I am trying to solve the system of equations below in matrix form.

A matrix is follows:

$\displaystyle \begin{pmatrix}3 &6 &-1 &-5 &4 &12 \\2& 4 &1 &0 &2 &5 \\1 &2 &-1 &-3 &1 &4 \\4& 8 & 0 &-4 & 2 & 8\end{pmatrix}$

After making reduced row echelon form operations on it I have:

$\displaystyle \begin{pmatrix}1 &2 &0 &-1 &0 &1 \\0 &0 &0 &0 &1 &2 \\0& 0 &1 &2 &0 &-1 \\0& 0 &0 &0 &0 &0\end{pmatrix}$

I have found that X_{1, }X_{3}and X_{5}are lead variables and that X_{2 }and X_{4 }are free variables:

If I let X_{2 }= s and X_{4 }=t. Then:

X_{5}= -2

X_{1}= 2-2s+t

X_{3}= -1-2s

Is this correct and is this how the answer should be presented?

Also, how can I find the basis for the U of $\displaystyle \mathbb{R}$^{4}from the matrix above?

Is this dimension of the U 3? and what is the dimension of U^{$\displaystyle \perp$}?

I would appreciate any help