N(A) and R(A) in terms of their basis

[DryRun]

Homework Statement

The matrix A =
1 1 1 1
-1 0 1 0
1 2 3 2

Express null space and row space of A in terms of their basis vectors.

2. The attempt at a solution

I have found the null space to be: x3 [1 -2 1 0]^T + x4 [0 -1 0 1]^T.

But my problem is how do i write the final answer correctly? Should i just write the answer as above? Or should i just write it this way: [1 -2 1 0]^T and [0 -1 0 1]^T

I did a search online and ended up with this way to present the solution, but there are so many variations, I'm confused.
{[1 -2 1 0]^T, [0 -1 0 1]^T}.

Which is the correct established answer format?

For the row space, i gave the answer like this:
[ 1 1 1 1] and [0 1 2 1]

Or should it be like this?: {[ 1 1 1 1], [0 1 2 1]}

 

[HallsofIvy]

The problem says to "Express the null space of A in terms of its basis vectors.
So I would say you can give the answer in either of two ways:
1) "All vectors in the null space of A are of the form $x_1\begin{bmatrix}1 \\ -2 \\ 1 \\ 0\end{bmatrix}+ x_2\begin{bmatrix}0 \\ -1 \\ 0 \\ 1\end{bmatrix}$"
or
2) "A basis for the null space is $\{\begin{bmatrix}1 \\ -2\\ 1\\ 0 \end{bmatrix}, \begin{bmatrix}0 \\ -1 \\ 0 \\ 1\end{bmatrix}\}$".

But the words explaining what you answer means are as important as your vectors.

 

[DryRun]

Thanks for your help, HallsofIvy. I will keep your advice in mind for my exams.

 

[HallsofIvy]

Yep- actually writing out full sentence answers is likely to send your teacher into shock!

 

[ahsanxr]

You can also say that the null space is span{(v1), (v2)}.