Can anyone verify that I am doing this correctly?

I have to find a basis for

So, does this mean that

and

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

, this gave me

. Then choosing

or

, I obtain

or

, respectively.

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

Repeating this process for a different

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.