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Linear Algebra
Step 1 of 22

The vector

Draw an arrow from the origin, ending two steps right and one step up.

Now write down (2,1)(2, 1).

Those are the same object. That is not a definition to accept; it is a choice somebody made, and the entire subject rests on it. Drag the arrow and watch the numbers follow. Nothing later works until this feels obvious.

Why the choice matters

Geometry is good at seeing. Look at an arrow and you instantly know it points up and to the right. But you cannot compute with a picture.

Arithmetic is good at calculating. You can add (2,1)(2,1) and (1,3)(1,3) in your head. But a column of numbers shows you nothing at a glance.

Fixing an origin and two directions gives you both at once. Every arrow gets an address; every address describes an arrow. A geometric question can now be answered by arithmetic, and an arithmetic answer can be checked by looking.

What the numbers actually say

(2,1)(2, 1) is not really "the point two right and one up". Read it as an instruction: take 2 steps along the first direction, then 1 along the second.

That reading looks pedantic now. Keep it anyway. In seven steps it becomes the reason matrices multiply the way they do, and skipping it is exactly why that rule feels arbitrary to most people forever.

Beyond the picture

Three entries, (2,1,5)(2, 1, 5), is an arrow in 3D. Still picturable.

784 entries is a 28×2828 \times 28 grayscale image, one number per pixel. Not picturable, and it does not matter: every rule ahead is built from the two-dimensional case and applies unchanged. That is exactly why the picture is worth the effort now.

One arrow alone cannot do much. Next: the only two things you are ever allowed to do to it.