In this question we denote by $\mathbb{P}_{2}\mathbb{R}$ is the set of functions {${ax^{2}+bx+c: a,b,c \epsilon \mathbb{R}}$}, which is a vector space under the usual addition and scalar multiplication of functions. Let $p_{1}, p_{2}, p_{3} \varepsilon$ $\mathbb{P}_{2}\mathbb{R}$ be given by $p_{1}(x)=1, p_{2}(x) =x+2x^{2}, and p_{3}(x) = \alpha x+4x^{2}$
(a) Find the condition on $\alpha \epsilon \mathbb{R}$ that ensures $p_{1}, p_{2}, p_{3}$ is a basis for $\mathbb{P}_{2}\mathbb{R}$. You are free to assume 1, x, $x^{2}$ are linearly dependent
(b) In the case $\alpha =5$, write the function $p(x)=1+x+x^{2}$ as a linear combination of $p_{1}, p_{2}, p_{3}$
Hi! For (a) i know for it to be a basis, it would have to be linearly independent and a span, so far i'm thinking $\alpha$ = 0... and to check span i get the equations from equating the variables...
2b-4c = $\alpha$,
b+$\alpha$c = $\beta$ and
a= $\gamma$
Pretty lost for both parts :( so would appreciate any hints or help on this question
Thanks!
(a) If you look at $p_2$ and $p_3$ you should see that $\{p_2, p_3\}$ is lin. independent $ \iff \alpha \ne 2$.
In this case it is easy to see that $\{p_1, p_2, p_3\}$ is lin. independent !
(b) If $p=ap_1+bp_2+cp_3$, then show that
$a=1, b+5c=1$ and $2b+4c=1$. It is your turn to solve this linear system !