Question: I'd really like someone to explain this slide to the class in their own words. Also, how does this slide related to the multiple coordinate systems described on slide 19?
dvernet
The slide is explaining how linear transformations can also be thought of as coordinate transformations -- in this case from a normalized space to one that can be represented on a W x H screen. Intuitively, what's going on here is that we have some linear transformation matrix $A$ that we've constructed as a way to map points in normalized space to the "perspective" of screen coordinates, according to a set of transformations.
Formally, the columns of $A$ form a basis for the coordinate space representing the W x H screen. So when we have a matrix vector equation such as $A\vec{x} = \vec{b}$, what's really going on is that we're taking the point $\vec{x}$ in normalized space, and mapping it to a point $\vec{b}$ in screen space by using the terms $x_{1}, x_{2}\ \in\ \vec{x}$ as weights in a linear combination of the columns of $A$:
$$
\begin{align}
A\vec{x} &=
\begin{bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{bmatrix}
\begin{bmatrix}
x_{1} \\
x_{2}
\end{bmatrix} \\
&=
x_{1}\cdot
\begin{bmatrix}
a_{11} \\
a_{21}
\end{bmatrix} +
x_{2}\cdot
\begin{bmatrix}
a_{12} \\
a_{22}
\end{bmatrix} \\
&= \vec{b}
\end{align}
$$
Question: I'd really like someone to explain this slide to the class in their own words. Also, how does this slide related to the multiple coordinate systems described on slide 19?
The slide is explaining how linear transformations can also be thought of as coordinate transformations -- in this case from a normalized space to one that can be represented on a W x H screen. Intuitively, what's going on here is that we have some linear transformation matrix $A$ that we've constructed as a way to map points in normalized space to the "perspective" of screen coordinates, according to a set of transformations.
Formally, the columns of $A$ form a basis for the coordinate space representing the W x H screen. So when we have a matrix vector equation such as $A\vec{x} = \vec{b}$, what's really going on is that we're taking the point $\vec{x}$ in normalized space, and mapping it to a point $\vec{b}$ in screen space by using the terms $x_{1}, x_{2}\ \in\ \vec{x}$ as weights in a linear combination of the columns of $A$: $$ \begin{align} A\vec{x} &= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} \\ &= x_{1}\cdot \begin{bmatrix} a_{11} \\ a_{21} \end{bmatrix} + x_{2}\cdot \begin{bmatrix} a_{12} \\ a_{22} \end{bmatrix} \\ &= \vec{b} \end{align} $$