Basis of the polynomial with degree less or equal 2

Question

Can u explain me one thing. We have $P_2:= \{\text{all polynomial with degree}\leq 2\}$ and $U_0:=\{f \in P_2 \mid f(1)=0\}$ We have $f(1)=c_1+c_2+c_3$ (because every polymial has form of $ f(t)=c_1 +c_2\cdot t +c_3\cdot t^2.$ So we have as basis of $U_0 =\begin{bmatrix} -1 \cr 1 \cr 0 \end{bmatrix},\begin{bmatrix} -1 \cr 0 \cr 1 \end{bmatrix} $. So we have {t-1, t^2-1}. My question is, how did we get this vectors? Thank u for your help!

Answer 1

You are looking at vectors $(c_1,c_2,c_3)^T$ where $c_1=-c_2-c_3$.

$$\begin{pmatrix} c_1 \\ c_2\\ c_3\end{pmatrix} = \begin{pmatrix} -c_2-c_3 \\ c_2\\ c_3\end{pmatrix} = c_2\begin{pmatrix} -1 \\ 1\\ 0\end{pmatrix}+c_3\begin{pmatrix} -1 \\ 0\\ 1\end{pmatrix}$$