Matrices as linear transformations

Drag $v$ anywhere. $Mv$ always stays exactly $90°$ ahead of it — the matrix isn't picking a new direction each time, it's applying the same fixed rule to whatever you hand it.

A $2\times2$ matrix is that fixed rule —

• $M$ — the $2\times2$ matrix representing the transformation.
• $a, b, c, d$ — the four entries of $M$ that determine what kind of transformation it performs.
• $v$ — the input vector being transformed, with components $v_x, v_y$.
• $Mv$ — the output vector after applying the transformation.

1. Each output is a weighted mix

   Each output component is a weighted mix of $v$'s own components.

2. The entries decide the transformation

   Change $a,b,c,d$ and you change what kind of transformation it is — a stretch, a rotation, a shear, or a projection that collapses a whole dimension away.

Switch matrices and drag $v$ around each time. The same arrow gets rotated, stretched, sheared, or flattened onto an axis depending only on which numbers are sitting in $M$.

With $M = \begin{pmatrix} 2 & 0 \\ 0 & 0.5 \end{pmatrix}$ (the "Scale" preset) and $v=(3,4)$:

1. Multiply the matrix by the vector

   $$Mv = \begin{pmatrix} 2(3) + 0(4) \\ 0(3) + 0.5(4) \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}$$
2. Read what each component did

   The $x$-component doubled, the $y$-component halved — exactly what a diagonal matrix does: scale each axis independently. Push that idea further — set the second diagonal entry to exactly $0$, as in the "Project" preset — and the $y$-component vanishes entirely: every vector collapses onto the $x$-axis. Projection is just a transformation with nothing left in one direction.