2×2 Matrix Times a Vector

For $M = \begin{pmatrix}a & b\\ c & d\end{pmatrix}$ and $\vec v = \begin{pmatrix}x\\ y\end{pmatrix}$, $M\vec v = \begin{pmatrix}ax + by\\ cx + dy\end{pmatrix}$. The columns of $M$ are exactly the images of the basis vectors $\hat e_1 = (1,0)$ and $\hat e_2 = (0,1)$.

Slide the four entries $a, b, c, d$. The lab shows the images of standard basis vectors $(1, 0)$, $(0, 1)$ and $(1, 1)$ under $M$. Try $a = d = 0$, $b = c = 1$: a swap matrix that flips $\hat e_1$ and $\hat e_2$.

Because any vector $\vec v = x\hat e_1 + y\hat e_2$, so $M\vec v = xM\hat e_1 + yM\hat e_2$ — a linear combination of the columns. This is the geometric heart of linear algebra: matrices are determined by what they do to a basis.

2-D rotation by angle $\theta$ uses the matrix $\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}$. Computer graphics, image transformations on Photoshop, robot arm kinematics, and even Markov chain transitions in two-state systems all use $2\times 2$ matrix-vector products.