I have a question and please I need help. First all, this is the context:
Let $p : T\rightarrow\mathbb{R}$ a polynomial of degree $\leq k$ with null measure condition onto T, that is $$\int_T p\ =\ 0,$$ where $T$ is some region in $\mathbb{R}^n$. I want to construct a basis of the subespacio where $p$ lies, namely $Q_{k,0}$.
Now suppose the I have $\{\varphi_0,\varphi_1,\ldots,\varphi_N\}$ an arbitrary basis for $P_k(T) = $ polynomial space of degree $\leq k$ onto $T$. Then, in order to obtain a basis for $Q_{k,0}$, I defined $$\psi_i\ =\ \varphi_i - \frac{1}{|T|}\int_T\varphi_i,\quad i = 0,1,\ldots,N.$$ Observe that $\{\psi_0,\psi_1,\ldots,\psi_N\}\subseteq Q_{k,0}$, but is not a basis, because they are linearly dependent.
So, my question is: how I can get a basis of $Q_{h,0}$ from the set $\{\psi_i\}$? In other words, how I know WHICH $\psi_i$ must be delete, to get a basis?
Remember, $\{\varphi_i\}$ is an arbitrary basis, because if we now that one $\varphi_i$ is constant, then the respective $\psi_i$ must be delete. The problem is, for example, for the Lagrange basis.
Thanks so much in advance for any hints.