suppose you have a vector field of second power polynomials, the the standard basis is $\{1,x,x^2\}$. however there is a method to constract another basis namely $\{1-x,1-x^2\}$.
1- how is it possiple to have 2 basis vectors? when dealing with second power polynomials the minimum vectors we need are 3 to span the space? 2- anyone knows this method?? im sure of the answer but not so sure how to get there.
oke so i figured out how to. i will post the answer for those who are intrested in learning more about algebra.
a second power polynomial can be written as follows: $$ p(x)=ax^{2}+bx+c $$ but the condition is given that $p(1)=0$, so then we get: $$ a+b+c=0 $$ and now do do a coordinate transformation with respectto the standard basis: $$ {[p]}_{E}= \begin{bmatrix} a \\ b \\ c \end{bmatrix} $$ now we must find $Nul(A)$ in other words $A{[p]_{E}}=\textbf{0}$. $$ {[p]}_{E}= \begin{bmatrix} -x_{2}-x_{3} \\ x_{2} \\ x_{3} \end{bmatrix}=-x_{2} \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} -x_{3} \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} $$ I hope you see that the last 2 vectors also span the space i.e. they form another basis $\{1-x,1-x^{2}\}$ with $dim(P)=2$!!