Learning Resources
- $\begin{bmatrix}\hat{i}_x & \hat{j}_x \\ \hat{i}_y & \hat{j}_y\end{bmatrix}
\begin{bmatrix}x \\ y\end{bmatrix}
= f\left( \begin{bmatrix}x \\ y\end{bmatrix} \right)$, where $\begin{bmatrix}\hat{i}_x \\ \hat{i}_y\end{bmatrix}$ is where the basis vector $\hat{i}$ lands after the transformation, and $\begin{bmatrix}\hat{j}_x \\ \hat{j}_y\end{bmatrix}$ is where the basis vector $\hat{j}$ lands.
- $det\left(\begin{bmatrix}v_x & u_x \\ v_y & u_y \end{bmatrix}\right)
=
\text{Area of parallelogram spanned by } \vec{v} \text{ and } \vec{u}$
- More accurately, the determinant is the increase in scale of the area (from the area of parallelogram formed by basis vectors $\hat{i}$ and $\hat{j}$). Since the basis vectors are unit vectors, that area is always 1.
- Matrix $A$ with $n$ columns will have a Rank $r$ (dimension of the span of all possible output vectors transformed by $A$) + nullity $X$, where $X=n-r$.
- $\text{rank}(A) + \text{nullity}(A) = n.$
- *Inaccurate:
All possible vectors in $n$-dimension space must either end up in the rank space or the nullspace.
- To get the nullspace, solve $Ax=0$, where $A$ is a transformation matrix, $x$ is an input vector, and $0$ is the zero vector.
- The null space is about “which inputs collapse.”
- The column space is about “which outputs are achievable.”
- https://www.youtube.com/watch?v=DA-iQkSSjKw
- Duality: $\begin{bmatrix}
a \\
b
\end{bmatrix}
\cdot
\begin{bmatrix}
c \\
d
\end{bmatrix}
=
\begin{bmatrix}
a & b
\end{bmatrix}
\begin{bmatrix}
c \\
d
\end{bmatrix}
= \text{A scalar value}$
- The Dot Product applies Linear Transformation $\begin{bmatrix}a & b\end{bmatrix}$ to an input vector $\begin{bmatrix}c \\ d\end{bmatrix}$ and outputs a scalar value (single number on the 1D number line).
- If you flip this transformation vector $\begin{bmatrix}a & b\end{bmatrix}$ on its head to become $\begin{bmatrix}
a \\
b
\end{bmatrix}$, you get the dual vector of this transformation in the original (pre-transformation) plane.
- $\vec{i}=\text{input vector, and } \vec{d}=\text{dual vector} \\ \vec{i} \cdot \vec{d} = X \\
\text{where } X= \text{the number on the number line (1D scalar) which the input vector lands on after the transformation}$
- The bottom right image shows this transformation (observe where the unit vectors - green and red arrows land)
