Some small linear algebra questions

[Raphisonfire]

Hello, I just have some small questions about linear algebra that are giving me some slight grief.

I know how to find the number of free variables and the basis of the solution space.

But what is confusing me is the A transpose... When doing the problem, would I firstly take the transpose of A and then row reduce the matrix to find the free variables? or would I do it the other way around?

Secondly..

I've solved part a)..

But I am stuck on part b) I have no idea where to start the problem or what to actually use to actually attempt solving this part. So a suggestion as to what I can actually do, would be very helpful!thank you in advance.

 

[Chaos2009]

For the last part, one idea that comes to mind is that b_1 \in W and b_2 \in W^{\perp} means that b_1 = a \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix} + b \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix} for some a,b \in \mathbb{R} and similarly b_2 would be a linear combination of the basis you found for W^{\perp}.

 

[Raphisonfire]

For the last part, one idea that comes to mind is that b_1 \in W and b_2 \in W^{\perp} means that b_1 = a \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix} + b \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix} for some a,b \in \mathbb{R} and similarly b_2 would be a linear combination of the basis you found for W^{\perp}.

Well I think I've solved it, finally... If I'm correct, I got

b = \begin{pmatrix} 2\\ 5 \\ 7 \\ 10 \end{pmatrix} = b_1 \begin{pmatrix} 8.1 \\ 10.2 \\ 12.3 \\ 14.4 \end{pmatrix} + b_2 \begin{pmatrix} -6.1 \\ -5.2 \\ -5.3 \\ -4.4 \end{pmatrix}

Also can anyone explain the first question to me? Thanks, in advance!