When you write a vector as \begin{bmatrix} 3 \\ 2 \end{bmatrix}, you are quietly using the standard basis: "3 steps along \hat{\imath}, 2 along \hat{\jmath}." The numbers 3 and 2 are the coordinates of the vector in that basis — the weights of the unique linear combination that reaches it.

Choose a different basis and the very same point gets different coordinates. The arrow in space hasn't moved — only the yardsticks we measure it against have changed.

Same point, new numbers

The black dot is fixed in place. Tilt the basis vectors \vec{b}_1 and \vec{b}_2, and read off how many of each it takes to reach the dot. The coordinates (c_1, c_2) change as the basis turns, even though the target never budges.

Why change basis at all?

Because the right basis can make a hard problem easy. Pick coordinates aligned with the grain of a problem and messy formulas turn simple. This is the secret behind diagonalization and behind principal component analysis, which hunts for the basis that best lines up with the spread of a dataset. Changing basis is just changing your point of view — and a good point of view is half the battle.

An example: line up with the data

Suppose three points happen to lie on the line y = x: (1,1), (2,2) and (3,3). In the standard basis each needs two numbers, and the bare pairs don't obviously shout "we are collinear".

Now measure against a basis aligned with the line — one vector pointing along it, one across it:

Every point on the line is (k, k) = k\,\vec{b}_1 + 0\,\vec{b}_2, so its coordinates in this basis are simply

The second coordinate is always zero. In the aligned basis the data is secretly one-dimensional — a single number now pins down each point, and that fact leaps off the page where the original (x, y) pairs hid it. The same move tames a tilted ellipse (align the axes with it and its equation collapses to \tfrac{x'^2}{a^2} + \tfrac{y'^2}{b^2} = 1), and finding the basis that flattens a whole cloud of data this way is exactly what principal component analysis does.