I am given the triangle $\Delta$ in $\mathbb{R}^2$ in which the vertices are given as the points $e_1=(0,0), e_2=(1,0)$ and $e_3=(0,1)$. The space $\mathbb{P}$ is defined as the set of bivariate polynomials of degree $\leq 1$.
In the first step I am asked to find a basis $S_1,S_2,S_3$ of the space $\mathbb{P}$ such that $S_i(e_j)=\delta_{i,j}$ holds for all $i,j=1,2,3$ .
Since each $S_i$ has to be of the form $\alpha+\beta \xi_1 +\gamma \xi_2$ we can just solve the following system of equations:
$\left ( \begin{array}{ccccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 &1 \end{array}\right ) \left( \begin{array}{c}\alpha \\ \beta \\ \gamma \end{array}\right ) = \left( \begin{array}{c} 1 \\0 \\0\end{array} \right);\left( \begin{array}{c} 0 \\1 \\0\end{array} \right); \left( \begin{array}{c} 0 \\0 \\1\end{array} \right) $.
The vectors on the right side correspond to $S_1,S_2,S_3$ respectively. That gives me the following solution:
$S_1(\xi_1,\xi_2)=1-\xi_1-\xi_2$
$S_2(\xi_1,\xi_2)=\xi_1$
$S_3(\xi_2,\xi_2)=\xi_2$.
In the next step I would like find basis functions $\phi_1,\phi_2,\phi_3$ for arbitrary vertices $x_1,x_2,x_3$ of the triangle $\mathbb{T}$, i.e. find $\phi_1,\phi_2,\phi_3$ such that $\phi_i(x_j)=\delta_{i,j}$. I was thinking about trying to generalize the result from above using the coordinate transformation
$\Delta \rightarrow \mathbb{P}: x=(\xi_1,\xi_2) \mapsto x_1+\xi_1(x_2-x_1)+\xi_2(x_3-x_1)$.
Is this the right idea? I'm having trouble to rewrite the functions $S_1,S_2,S_3$ in the new coordinate system. Could someone help me with that?
Also, I'm asked to determine the mass matrix $M$ given by \begin{equation} M_{i,j}=\int_{\mathbb{T}} \phi_i \phi_j dx \end{equation} Here I had the same idea as above. I first determined the mass matrix for the fuctions $S_1,S_2,S_3$ which gave: $M_{S_1,S_2,S_3}=\left ( \begin{array}{ccc} 1/12 & 1/24 & 1/24 \\ 1/24 & 1/12 & 1/24 \\ 1/24 & 1/24 &1/12 \end{array}\right ) $
I wonder if there is a way to generalize this result to the functions $\phi_1,\phi_2,\phi_3$ using coordinate transformation?
