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**paupsers** Can anyone verify that I am doing this correctly?

I have to find a basis for

$\displaystyle W=\{(a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \in F^{5} | a_{1}-a_{3}-a_{4}=0\}$

So, does this mean that $\displaystyle a_{2}$ and $\displaystyle a_{5}$ can be any numbers? **yes!**

I began by setting them both equal to one, giving me (0, 1, 0, 0 ,0) and (0, 0, 0, 0, 1)

Then, by letting $\displaystyle a_{1}=1$, this gave me $\displaystyle 1=a_{3}+a_{4}$. Then choosing $\displaystyle a_{3}=0$ or $\displaystyle a_{3}=1$, I obtain $\displaystyle a_{4}=1$ or $\displaystyle a_{4}=0$, respectively.

So that gave me the vectors (1, 0, 0, 1, 0) and (1, 0, 1, 0, 0).

Repeating this process for a different $\displaystyle a$ gave me

(0, 0, 1, 1, 0).

That's a total of 5 vectors, which is how many the basis should have, correct? Can anyone verify this solution? Is this a correct method to use? I just feel weird "choosing" the a's to be 1 or 0 and stuff.