Linearity Is Permission to Break Things Apart
We've looked at matrices, determinants, and eigenvalues — but there's a single property underneath that makes all of it run. Linearity. Grand name, one meaning: "break it into pieces, handle them separately, add them back, and you get the same answer as doing it all at once." It's the promise that you can split a hard thing into simple parts and solve them. Lesson 2 said every vector is a combination of î and ĵ; lesson 5 said a matrix only needs to track î and ĵ. The reason that works is exactly this linearity. This lesson is the bridge between those two.
For a transformation to be "linear," it just has to obey two rules. One: adding then transforming gives the same result as transforming then adding. Two: scaling up then transforming gives the same result as transforming then scaling up. Check it right here. Drag the vectors a and b and watch the two paths always land on the same spot. Obey those two rules and the transformation is linear. Seems like nothing, right? But this "nothing" of a promise is what makes everything else possible.
Take the first rule in detail. One path: add a and b first (a+b), then transform. Other path: transform a and b separately, then add the results. The two paths land in exactly the same place. It doesn't matter whether you add first or transform first. Move a and b around. The two paths always meet at the same point. So you never have to handle the combined thing as a lump — you can handle the parts separately.
The second rule. Stretch v by 3 then transform it, versus transform v then stretch by 3. Same again. Double the input and the output doubles exactly; triple it and it triples. Cleanly proportional, no extra. Drag the scale factor on the slider. Whether you stretch before or after, you get the same result. What you put into the input shows up untouched in the output — that's the core feel of linear.
Put the two rules together and something huge falls out. Any vector can be split into an î piece and a ĵ piece, just like in lesson 2. Thanks to linearity, the transformation acts on each piece on its own, and you just add the results. So transform(any vector) = (its î-amount)·transform(î) + (its ĵ-amount)·transform(ĵ). Know only where the transformation sends î and ĵ, and every one of the infinitely many other vectors comes along for free. Drag a vector and watch it split into î and ĵ pieces, get transformed, and recombine. This is exactly why lesson 5's matrix only needed its two columns, where î and ĵ go — linearity covers the rest.
This rule has a famous name in engineering: superposition. If input A makes output A and input B makes output B, then input A+B makes exactly output A+B. Combining throws up no surprises. So you take a messy input, break it into simple pieces, solve each one, and add the answers. Splitting a sound into pure tones (coming up in Fourier), computing each power source in a circuit separately and adding — all of it is superposition. Nonlinear things don't do this. Just look at squaring: (a+b)² isn't a²+b². Combine, and something unexpected pops out. That's why almost every problem we can solve cleanly is a linear one.