[cygnus_a]

The TLDR intuition of a change of basis is this: you have a vector of length L that points from the origin up towards the z-axis. If you'd like to rotate your perpendicular measuring sticks like x->x', y->y' and z->z' (ie, rotate your coordinate system, maintaining perpendicular-ness and vector length), then you've changed your basis. Say you rotate your coordinate system 90 degrees about the y-axis, now your vector points in the new x-direction (so x-> -z, y->y and z->x, and your vector is (1,0,0) instead of (0,0,1)).

You can do that with a matrix operation (M*v = v'):

[0 0 1]___[0__ __[1

[0 1 0]___0__=__0

[-1 0 0]__1]__ __0]

In fact, this matrix rotates any vector 90 degrees about the y-axis.

Basis vectors can also be more abstract than that. For instance, they're useful in quantum mechanics for simplifying the schroedinger eq (sometimes from a second order diff eq to a first order one) by changing from a position basis to a momentum basis, in effect rewriting your derivatives from a different point of view.

[j2kun]

> rotate your coordinate system, maintaining perpendicular-ness and vector length

Not necessary for every change of basis.

[cygnus_a]

Yes, it has to be a normalized orthogonal basis to maintain those properties :)