I would like to check if my understanding of projection maps is correct.
I have been given the following subset of $\mathbb{R}^3$:
$A=\left\{\begin{pmatrix} x \\ y \\ -x+2y \end{pmatrix} \middle| x,y,z\in\mathbb{R}\right\}$
A basis for this subset is $\mathscr{B}=\left\{ \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix},\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} \right\}$, and to extend this basis to one for the vector space $\mathbb{R^3}$ we simply add to the basis the vector: $\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$
To obtain $\mathscr{C} = \left\{ \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix},\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix},\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \right\}$, a basis for $\mathbb{R}^3$.
We can call $B = Span\left\{\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\right\}$, and then we can say $\mathbb{R}^3=A\bigoplus B$.
What I want to know is if I am correct in interpreting the definition of projection map. Let $P:\mathbb{R}\to\mathbb{R}$ be the projection map onto A. The question asks me to calculate $P(e_1)$, $P(e_2)$ and $P(e_3)$ then write down the matrix of $P$ with respect to the standard basis of $\mathbb{R}^3$. Without explicitly giving my answer (I want to check my method, not my answers), this is my method:
Write each vector $e_1$, $e_2$ and $e_3$ as a linear combination of the vectors in $\mathscr{C}$, so, for example, $e_1 = \alpha\begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}+\beta\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} \gamma\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$.
For the projection map onto $A$ we take only the first two terms as the first two terms are in the basis $\mathscr{B}$. So, for the combination in step 1, $P(e_1)=\begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix} = \alpha e_1+\beta e_2+\gamma e_3$
To form the matrix P we write down the columns of the matrix the coefficients describe in the last step, so we get: $P=\begin{pmatrix} \alpha & . & . \\ \beta & . & . \\ \gamma & . & . \end{pmatrix}$, and fill in the missing columns as we did for the first column above.
Am I correct in my method? If I have any of this wrong, please guide me in the right direction.