Coordinates Are a Point of View
Point at one building and someone says "third floor, red door," another says "two doors left of the entrance." Same spot, different numbers. Vectors are like that. The arrow stuck in space is one fixed thing, but the coordinates that read off its position depend on the ruler we chose — the basis. Change the basis and the arrow doesn't budge; only the numbers change. And one well-chosen basis can make a transformation that looked tangled suddenly look simple. This is the secret behind how AI handles data: a "feature" is, in the end, a coordinate read in a basis we picked.
The blue arrow P is nailed to the same spot on screen. Use the button to switch the ruler between the standard basis and a skewed basis. In the standard basis the grid is square and P reads as (2.0, 1.0). Flip to the skewed basis and the grid tilts along green b1 and orange b2, yet the P arrow doesn't move a millimeter. All that changes is the numbers read off it. Now P is written as so many b1 plus so many b2. The arrow is, in truth, one thing sitting in place no matter who names it or how.
Now drag the blue vector v yourself. As you drag, two pairs of numbers move together. On top are the standard coordinates (vx, vy); below are (α, β) read in the skewed basis. Same arrow, translated into two languages at once. Follow the skewed grid lines and you see it directly: α is how many steps along b1, β how many along b2. When v happens to line up with b1, β nears zero; line it up with b2 and α nears zero. A coordinate is nothing more than the arrow measured through that basis's eyes.
You can pack a basis into a single matrix: stand the new basis vectors up as columns to make B = [b1 b2]. Move B's four entries with the sliders and the stage shows the standard grid bending along B. Here's the key: the columns of B are where the standard basis vectors land — that is, they are the new basis vectors. And the reverse job is B's inverse, which turns standard coordinates into new-basis coordinates. Ease the sliders until det nears zero and the inverse readout shows "singular": the basis has flattened onto a line and can no longer cover the plane.
If a coordinate unpacks an arrow, then in reverse you can assemble an arrow from coordinates. Move the sliders α and β. First draw the green arrow α·b1 from the origin, then attach the orange arrow β·b2 from its tip — tip-to-tail, the parallelogram addition. The result is the blue v. However you set α and β, every point of the plane is built in exactly one way: none missed, none doubled. That is why (α, β) becomes v's true address in that basis.
Now for the payoff. Fix one transformation A and use the button to flip between two viewpoints. In the standard view the grid warps and an ordinary arrow even twists in direction; it's hard to see at a glance what's going on. Flip to the eigenbasis view, which stands A's two eigenvectors up as the new axes. Then A becomes a pure stretch along those two axes: each arrow only changes length, it does not rotate. Same transformation, but seen in the right basis it has gone as simple as a diagonal matrix. The "directions that don't turn" from the eigenvectors lesson were exactly this good basis.