Basis
A set of reference directions that turns a vector into a list of numbers — change the directions and the numbers change, though the vector does not.
Basis
A basis is a minimal set of vectors whose linear combinations reach every point in a space, with none of them redundant. In the plane, any two non-parallel vectors form a basis. Choosing a basis is what lets you replace an abstract arrow with a concrete list of coordinates: the numbers tell you how much of each basis vector to add up. The crucial subtlety is that the coordinates are not the vector — they are the vector as seen from a particular set of axes.
Same vector, different numbers
The standard basis \hat\imath = (1,0), \hat\jmath = (0,1) feels canonical only by habit. The very same arrow has different coordinates in a tilted basis. Below, one fixed vector (white) is read off in two coordinate systems — the standard grid and a rotated one. The arrow never moves; only the numbers describing it do.
Change of basis
Switching bases is itself a Linear Transformation. If the new basis vectors are the columns of a matrix P, then P converts new coordinates into old, and P^{-1} converts old into new. When the basis is orthonormal — mutually perpendicular unit vectors — the conversion is delightfully cheap: each coordinate is just the Dot Product of the vector with the corresponding axis, no matrix inverse required.