Change of Basis

A vector is a thing in the world, an arrow, and it does not care how you describe it. Coordinates are the description, and they depend entirely on the basis you measure against. Pick two basis vectors $b_1$ and $b_2$ and the same arrow $v$ acquires new coordinates $(c_1, c_2)$, the amounts of each basis vector you stack to reach it: $v = c_1 b_1 + c_2 b_2$. Collect the basis vectors as the columns of $P = [\,b_1\ \ b_2\,]$ and this reads $Pc = v$, so the coordinates are $c = P^{-1}v$, while the familiar standard basis just returns $c = v$. The top panel draws both readings at once: the faint square grid of the standard basis and the slanted grid of yours, with $v$ resolved along the oblique axes into its two components. Drag $b_1$ and $b_2$ to re-grid the plane and the coordinates of the unmoved vector change before your eyes, the determinant of $P$ measuring how much the basis cell is stretched and sheared. The bottom panel sweeps $v$ once around a circle and plots its coordinates in each basis: clean cosine and sine in the standard one, the same circle re-read as phase-shifted, rescaled sinusoids in yours. This relabelling is the engine behind diagonalization, where the right basis turns a tangled matrix $P^{-1}AP$ into a diagonal one.