Background: I have a 2-dimensional 2-class classification problem, the following training sample vectors are given:
$$D_1 = \left\{ \begin{bmatrix} 0 \\ -1 \end{bmatrix}, \begin{bmatrix} 3 \\ -2\end{bmatrix}, \begin{bmatrix} 0 \\ 2 \end{bmatrix}, \begin{bmatrix} -2 \\ 1\end{bmatrix}, \begin{bmatrix} 2 \\ -1 \end{bmatrix} \right\} $$ and $$ D_2=\left\{ \begin{bmatrix} 6 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 2\end{bmatrix}, \begin{bmatrix} 9 \\ 1 \end{bmatrix}, \begin{bmatrix} 7 \\ 4\end{bmatrix}, \begin{bmatrix} 5 \\ 5 \end{bmatrix} \right\}$$
I used Fisher linear discriminant analysis method to calculate a transform vector $w$.
I wrote the code and I got
w =
-0.2308
-0.2033
Question: How can I calculate the projections of all the data samples in the resulting 1-dimensional space?
Do I just multiply the data with the $w$ and get new $D_1$ and $D_2$?
I found a slide, can someone explain this to me?
The projection from a $d$-D to $(c\!-\!1)$-D can be expressed as
$y_i = w_i^t x \quad i=1, \dots, c-1 \quad \text{or}$
$y=W^t x$, where $w_i$ are columns of $d \times (c\!-\!1)$ matrix $W$.