Change of Basis

What to observe

1. Both panels show the same arrow. On the left it is measured against our square grid, on the right against theirs, which we have redrawn as if it were the square one, because that is how it looks from inside. The arrow never moved. Only the numbers written under it did.
2. Drag v and watch the two coordinate readouts disagree. Then dragb₁ instead: v stays exactly where it is on the left, yet its coordinates on the right change completely. Coordinates are a statement about the pair (vector, basis), never about the vector alone.
3. The columns of P are just b₁ and b₂ written in our coordinates, so P takes their numbers and returns ours: v = P[v]_B. It reads in their language, and it is easy to remember which way round it goes because the columns are literally where their basis vectors are. Going the other way needs P⁻¹.
4. Pick the Degenerate preset. b₁ and b₂ line up, det P is zero, and the right-hand grid collapses. Two vectors that do not span the plane cannot describe every point of it, so there is no dictionary to invert:a basis has to be independent or nothing works.
5. Switch on Stretch b₁ ×2. In their grid the map is as simple as a map gets: multiply the first coordinate by two and leave the second alone, a diagonal matrix. Look at what the very same map has to be written as in our grid and it is a mess of four unrelated numbers.
6. That mess is A = P A′ P⁻¹, and reading it right to left tells you exactly what it does: translate our numbers into theirs, do the easy thing there, translate back. Diagonalising a matrix is nothing more than hunting for the basis in which it was easy all along, and the eigenvectors are that basis.
7. Rotate the basis with the Rotated preset and check det P. When the new basis is orthonormal, det P is 1, lengths and angles survive the translation, and P⁻¹ is simply Pᵀ. That is the case worth remembering, and the reason orthonormal bases are worth the trouble of building.