Linear Transformations

A linear transformation is simply a rule that takes every vector and moves it somewhere else, in a way that plays fair with adding and scaling. It's called linear because it keeps straight lines straight (at worst it squashes one down to a single point), and transformation is just the word for a function that moves vectors around. It's good for describing rotations, reflections, stretches and shears, which is everything a graphics engine does to a 3D model. The two rules above have just a few pieces: $T$ (the rule itself, which eats a vector and returns a vector), $T(\vec u + \vec v) = T(\vec u) + T(\vec v)$ (it doesn't matter whether you add before or after moving) and $T(c\,\vec u) = c\,T(\vec u)$ (it doesn't matter whether you scale before or after). It sounds abstract, but it's two fair-play rules; bear with it.

You've already seen one. Matrix multiplication from earlier in this unit satisfies exactly these two rules: $A(\vec u + \vec v) = A\vec u + A\vec v$ and $A(c\vec u) = cA\vec u$. So every matrix defines a linear transformation, $T(\vec x) = A\vec x$. In the next lesson you'll see that it works the other way too: every linear transformation from $\mathbb R^n$ to $\mathbb R^m$ comes from a matrix.

Watch the grid. Apply a linear transformation to the whole plane: grid lines stay parallel and evenly spaced, and the origin stays put. Rotations, reflections, stretches and shears are all linear; shifting (translation) is not, because it moves the origin. There's an animation further down that shows a shear and a rotation doing this to a grid, and after it a rotation you can control yourself.

A common mistake. It's tempting to think a shift like $T(x, y) = (x + 1, y)$ is linear, because its formula only has plain first-power terms, just like a linear equation. It isn't: a linear transformation always sends $\vec 0$ to $\vec 0$ (use $c = 0$ in the scaling rule: $T(\vec 0) = T(0\,\vec u) = 0\,T(\vec u) = \vec 0$), and this one sends $(0, 0)$ to $(1, 0)$. So a quick first check for any transformation is to see where the origin goes.

Linearity is powerful. If you know where $T$ sends a few vectors, you know where it sends all their combinations: $T(3\vec u - 2\vec v) = 3T(\vec u) - 2T(\vec v)$. That's the idea behind the matrix of a transformation, coming up next lesson.

Domain and codomain. An $m\times n$ matrix sends $\mathbb R^n$ to $\mathbb R^m$. The order trips people up, but think about it: $A\vec x$ only makes sense when $\vec x$ has one entry per column, so inputs have $n$ entries, and the output has one entry per row, so $m$ entries. The range is the set of all outputs: the span of the columns, since every output $A\vec x$ is a combination of them.

Graphics engines move, scale and rotate every vertex of every model with linear transformations each frame; image filters and audio effects are linear transformations of pixels and samples.