change of basis explained
For the same building, one person says "the red door on the 3rd floor" and another says "two doors left of the main entrance." They point to the same place but say different numbers. Vectors are like that too. The arrow fixed in space is one thing, but the coordinates that read its position depend on the measuring frame we choose, the basis. Change the frame and the arrow does not budge; only the numbers change. And one well-chosen frame can make a transformation that looked complicated simple in an instant. This is also the secret of how AI handles data, because a feature is, in the end, a coordinate read in a basis we chose.
The blue arrow P is nailed to the same spot on the screen. Use the button to switch the measuring frame between the standard basis and a skewed basis. In the standard basis the grid is square and the arrow reads (2.0, 1.0). Switch to the skewed basis and the grid tilts along the green b1 and the orange b2, but the arrow P does not move by even 1mm. Only the numbers read off it change, and the equation below now writes the arrow as so many b1 plus so many b2. However anyone names it, the arrow is the single thing that stays put.
This time, drag the blue vector v yourself. As you drag, two pairs of numbers move together: on top the standard coordinates (vx, vy), and below the coordinates (α, β) read in the skewed basis. It is one arrow interpreted into two languages at once. Follow the skewed grid lines and you can see directly that α counts steps along b1 and β counts steps along b2. When v happens to line up with b1, β approaches 0; when it lines up with b2, α approaches 0. A coordinate is nothing more than the arrow measured through the eyes of that basis.
A basis can be packed into a single matrix: stand the new basis vectors up as columns to form B = [b1 b2]. Move the four entries of the matrix with the sliders and the stage shows the standard grid deforming to follow it. The key point is that the columns of the matrix are where the standard basis vectors land, that is, the new basis vectors. The reverse job is done by the inverse matrix, which turns standard coordinates into new-basis coordinates. The inverse and the determinant are shown together below. Ease the sliders until the determinant approaches 0 and the inverse readout shows "singular." That is the moment the basis collapses onto a line and can no longer cover the plane. Press "Reset" to return to the standard basis.
If coordinates unpack an arrow, you can also go the other way and assemble an arrow from coordinates. Move the α and β sliders: first the green arrow α·b1 is drawn from the origin, and from its tip the orange arrow β·b2 is attached. It is tip-to-tail parallelogram addition, and the result is the blue arrow. However you set α and β, every point of the plane is built in exactly one way. No point is missed and none is made twice. That is why (α, β) is the true address of the arrow in that basis. The bottom line shows the standard coordinates of the assembled arrow.
Now comes the reward. Fix one transformation A and use the button to switch between two viewpoints. In the standard view the grid is twisted and an ordinary arrow even changes direction, so what is happening is hard to take in at a glance. Switch to the eigenbasis view and the two eigenvectors of the transformation become the new axes. Then the transformation is a pure stretch, lengthening and shortening along those two axes, and the two arrows do not turn; only their lengths change. The factor beside each arrow is that axis's eigenvalue, and both eigenvalues are also shown below. It is the same transformation, but seen in the right basis it has become as simple as a diagonal matrix. The directions that did not turn in "Eigenvectors Are the Directions That Don't Turn" were exactly this good basis.