Geometric Transformations in ℝ²

A geometric transformation of the plane is a rule that takes each point and moves it somewhere new. We write points as column vectors:

$\mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix},$

where $x$ and $y$ are the horizontal and vertical coordinates.

The transformations we care about here are linear: they keep the origin fixed, send straight lines to straight lines, and keep grid lines evenly spaced. Every such transformation $T$ can be written as multiplication by a fixed $2\times 2$ matrix $A$:

$T(\mathbf{v}) = A\mathbf{v}.$

If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the product is computed row-by-column:

$A\mathbf{v} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}.$

The image of $\mathbf{v}$ is the output vector $A\mathbf{v}$. So the whole geometric question — where does this point go? — becomes a single multiplication.

Here is the key idea that makes every matrix below easy to remember. Let the standard basis vectors be

$\mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \qquad \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix},$

the unit arrows pointing right and up. Multiplying $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ by them picks out the columns:

$A\mathbf{e}_1 = \begin{bmatrix} a \\ c \end{bmatrix}, \qquad A\mathbf{e}_2 = \begin{bmatrix} b \\ d \end{bmatrix}.$

So the first column of $A$ is where $\mathbf{e}_1$ lands, and the second column is where $\mathbf{e}_2$ lands. To build the matrix for any transformation, just ask two questions: Where does the right-pointing arrow go? and Where does the up-pointing arrow go? Stack those two answers as columns and you are done. To read a matrix, do the reverse: look at its columns to see what happens to the two basis arrows.

Using the column idea, here are the standard transformations. In each case $s_x, s_y, \theta, k$ are given numbers and $(x,y)$ is a general point.

Scaling by factors $s_x$ horizontally and $s_y$ vertically stretches each axis independently: $\operatorname{diag}(s_x, s_y) = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}, \qquad (x,y) \mapsto (s_x x,\; s_y y).$

Rotation counterclockwise about the origin by angle $\theta$: $R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}.$ For the clean angles: $R(90^\circ) = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$, $R(180^\circ) = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$, $R(270^\circ) = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$. For $45^\circ$, use $\cos 45^\circ = \sin 45^\circ \approx 0.7071$.

Reflections flip the plane across a line through the origin: $\text{across the } x\text{-axis: } \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \quad \text{across the } y\text{-axis: } \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}, \quad \text{across } y=x: \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}.$ Reflecting across $y=x$ simply swaps the coordinates: $(x,y)\mapsto(y,x)$.

Shears slide the plane parallel to one axis by an amount proportional to the other coordinate: $\text{horizontal: } \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix} \;\;(x,y)\mapsto(x+ky,\;y), \qquad \text{vertical: } \begin{bmatrix} 1 & 0 \\ k & 1 \end{bmatrix} \;\;(x,y)\mapsto(x,\;kx+y).$ Here $k$ is the shear factor controlling how much the plane is slanted.

To apply one transformation and then another, you multiply the matrices. If $T_1(\mathbf{v}) = A_1\mathbf{v}$ is done first and $T_2(\mathbf{v}) = A_2\mathbf{v}$ second, the combined effect is

$T_2\big(T_1(\mathbf{v})\big) = A_2\,(A_1\mathbf{v}) = (A_2 A_1)\,\mathbf{v}.$

Notice the matrix of the first transformation sits on the right, next to the vector — because that is what touches $\mathbf{v}$ first.

Order matters. Matrix multiplication is generally not commutative, $A_2 A_1 \neq A_1 A_2$, so rotating then reflecting usually gives a different result than reflecting then rotating. Always compose in the order the transformations actually happen.