Linear independence of polynomial set.

[bamuelsanks]

Hi guys,

I've been working on a question which is as follows:

For which real values of c will the set $\{1+cx, 1+cx^2, x-x^2\}$ be a basis for $P_2$?

I'm coming up with the answer as no values of c, but am I really wrong?
I've only checked linear independence, because it would imply that it spans $P_2$ (right?)

I figure one would just create the augmented matrix:
$\left( \begin{array}{ccc} 0 & c & 1 \\ 1 & 0 & 1 \\-1 & 1 & 0\end{array} \right)$

And reduce:
$\left( \begin{array}{ccc} 1 & 0 & \frac{1}{c} \\ 0 & 1 & \frac{1}{c} \\ 0 & 0 & 0\end{array} \right)$

Thanks in advance,
SB

 

[Amer]

for any c real number
$$1+cx = 1+cx^2 + c(x-x^2 )$$

that means the first one can be written from the other two elements (linear compination from other elements)thats means they can't be a basis.
other way to prove that they are basis or not try to generate the standard basis {1 , x , x^2 } from the given basis if that can be done then they are basis