If I have a vector $y_1 = (1,2,3)$ and $y_2 = (4,5,6)$, how to calculate $\prod\{y_{1}\}$ and $\prod\{(y_{i})_{1\leq i \leq2}\}$ according to the definition below?
denote $\Pi \{ y_1,...,y_k\}$ the orthogonal projector onto the subspace spanned by the vectors $y_1,...,y_k$. The projector is a $T \times T$ matrix, where $T$ is the length of these vectors.
Given a matrix $A$, the orthogonal projector $P$ onto the span of the columns of $A$ is given by, $$ P = A(A^*A)^{-1}A^*$$
Therefore, in your notation, $$ \prod\{y_1\} = \left[\begin{array}{c}1\\2\\3\end{array}\right] \left( \left[\begin{array}{c}1\\2\\3\end{array}\right]^*\left[\begin{array}{c}1\\2\\3\end{array}\right]\right)^{-1}\left[\begin{array}{c}1\\2\\3\end{array}\right]^* = \frac{1}{14}\left[ \begin{array}{ccc} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \\ \end{array} \right]$$ and $$ \prod\{y_1,y_2\} = \left[\begin{array}{c}1&4\\2&5\\3&6\end{array}\right] \left( \left[\begin{array}{c}1&4\\2&5\\3&6\end{array}\right]^*\left[\begin{array}{c}1&4\\2&5\\3&6\end{array}\right]\right)^{-1}\left[\begin{array}{c}1&4\\2&5\\3&6\end{array}\right]^* = \frac{1}{6}\left[ \begin{array}{ccc} 5 & 2 & -1 \\ 2 & 2 & 2 \\ -1 & 2 & 5 \\ \end{array} \right] $$