What is a Transformation?

You've learned what vectors are. Now imagine taking every single vector in the plane — every arrow starting from the origin — and moving them all at once according to some rule. That's a linear transformation.

Think of it as moving space

A transformation doesn't just move individual arrows. It moves the entire coordinate grid — every point, every line, every vector. Watch the green grid in the interactive as you try different transformations.

Notice what stays the same regardless of which transformation you pick:

• Grid lines remain straight — they don't curve or bend
• Parallel lines remain parallel
• The origin stays fixed — it never moves

These three properties are exactly what makes a transformation linear. Any function that preserves straight lines and keeps the origin in place is linear.

The key insight: just track the basis vectors

Here's the remarkable thing. Because the transformation is linear, you only need to know where two special vectors go — and everything else follows automatically.

Those two special vectors are $\hat{\imath} = [1,\ 0]$ (one step right) and $\hat{\jmath} = [0,\ 1]$ (one step up). They're called the basis vectors.

Any vector $[x,\ y]$ can be written as $x\,\hat{\imath} + y\,\hat{\jmath}$. So once you know where $\hat{\imath}$ and $\hat{\jmath}$ land, linearity tells you where every vector lands.

What doesn't count as a linear transformation

Translations (shifting everything sideways) are not linear — they move the origin. Squaring coordinates is not linear — it curves straight lines. Anything that bends lines or moves the origin is out.

Why matrices

Since you only need to record where $\hat{\imath}$ and $\hat{\jmath}$ land, you can store an entire transformation in just four numbers — the two components of each destination. That's exactly what a matrix is: a compact description of a linear transformation. We'll build that connection in the next lesson.