I'm researching a potential algorithm, and I'm hoping that someone can verify my calculations.
I have sets of vectors in $\mathbb{R}^6$ that I can use. They have a corresponding value associated with them, but this relation is not necessarily a simple one. I'd like to find the value associated with the vector $(0,0,1,0,0,0)$. So I'm wondering if I can perform linear algebra, using the vectors that I can create, to do so.
One vector I can create is $(a,a,a,0,0,0)$ for any real $a$. A second is $(b,b,b,b,b,b)$.
I can also create vectors of the form $(c^0,c^1,c^2,c^0,c^1,c^2)$ for some real number $c$, where $c^k$ is just simply $c$ taken to the $k$th power. A second form that I can create is $(0, d, 2d, 0, 0, 0)$ for real $d$.
I am wondering if these vectors form a complete basis for $\mathbb{R}^6$.
In case it helps, I can use as many vectors as I want, adding and/or subtracting them, as long as they are of the forms above.
My Question
What I really want to know is, can I find the corresponding value for $(0,0,1,0,0,0)$?