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Linear Algebra
Step 7 of 22

Transformations

Understand first: span and basis

Until now every vector was a thing you looked at. Now make them all move together.

A transformation takes every vector in and returns a vector out. Same idea as a function; the word changes because the mental image should. Do not picture a graph. Picture the entire plane picked up and stretched, with every point riding along.

Drag the two arrow tips and watch the whole grid follow.

The two rules that make it linear

Most transformations are chaos. A tiny family is well behaved, and only that family is worth studying. A transformation is linear when it keeps both promises:

T(u+v)=T(u)+T(v),T(c u)=c T(u)T(u + v) = T(u) + T(v), \qquad T(c,u) = c,T(u)

In words: it does not matter whether you add first and transform after, or transform first and add after. Same for scaling.

You can spot it without algebra. Drag the tips as far as you like and check the grid: lines stay straight, parallel lines stay parallel, spacing stays even, and the origin never moves. Bend any of those and linearity is broken.

Why that buys everything

Here is the consequence that makes the whole subject possible.

Every vector is a recipe: (2,1)(2,1) means 2ı^+1ȷ^2\hat{\imath} + 1\hat{\jmath}. Apply a linear transformation and the rules let you push TT through the recipe:

T(2ı^+1ȷ^)=2 T(ı^)+1 T(ȷ^)T(2\hat{\imath} + 1\hat{\jmath}) = 2,T(\hat{\imath}) + 1,T(\hat{\jmath})

Read what that says. To know where any of the infinitely many points in the plane goes, you need only two facts: where ı^\hat{\imath} lands, and where ȷ^\hat{\jmath} lands. Everything else follows from the recipe it was already carrying.

Infinite information, compressed to two arrows.

That compression deserves a container, and it has one. Next step: writing those two landing spots down, and discovering that the matrix rule you were taught was never arbitrary.