Finding matrices which P is null space

Question

Let P is set all vectors $(x_1,x_2,x_3,x_4) \in R^{4}$ for which $x_1 + x_2 + x_3 + x_4=0$

a) Find one base of subspace $P^{\bot}$

b) Prove that P is subspace of $R^{4}$ then construct matrices which P is null space.

a) I found base $(1,1,1,1)$ that is easy b) I know how to prove that $P$ is subspace, but I do not know if my matrix is good, because what means that P is null space, does that mean that for every vector $x\in P$ $Ax=0$. So matrices $A$ are

A=$\begin{bmatrix} 1& 1& 1& 1\\ 1& 1& 1& 1\\ 1 & 1& 1& 1\\ 1 & 1& 1& 1 \end{bmatrix}$.

Answer 1

Your answers are fine, but I don't know if you are expected to find just one matrix whose null space is $P$. The question says 'matrices'.

Answer 2

a) The vector $(1,1,1,1)$ is indeed the basis vector for $P^{\bot}$. Just a warning that the question may implicitly also require you to prove your claim, which is not difficult in this particular case, but may cause loss of points.

b) Your matrix is ok, but you could also just have the matrix $[1,1,1,1]$ which has the same nullspace.

Answer 3

Yes that's correct, a basis for $P$ is $(1,-1,0,0),(1,0,-1,0),(1,0,0,-1)$ and $(1,1,1,1)$ is a basis for $P^{\perp}$, for point "b" more in general matrices $A$ are n-by-4 matrices

$$\begin{bmatrix} a& a& a& a\\ a& a& a& a\\ \vdots& \vdots& \vdots& \vdots\\ a & a& a& a \end{bmatrix}$$

with $a \neq 0$.