TITLE: Why do we need "basis" in linear algebra?
QUESTION [5 upvotes]: I am very curios about why in linear algebra we need different basis, why can't we just have the standard basis and work with that?
And how basis is used in computer graphics?
thank you in advance.

REPLY [3 votes]: In computer graphics, changes of basis are frequently used to describe relative positions.  For example, say we want to model a 3D person.
Now, we could certainly model this by figuring out the absolute translation, shear, and rotation for each part of the body.  But that gets ugly very fast.  We need to do lots of work to figure out where each piece of the body should go so that it all looks natural.  And if we ever want to move any part of the body, we need to redo a lot of this calculation.
But what we usually want to talk about is the relative position.  We don't want to talk about the position of the head in terms of the origin, we want to describe it in terms of the neck.  To do so, we use a relative coordinate system, where the origin is at the neck.  Then we just need to describe it with one position, shear, and rotation.  And if the neck ever gets moved, the head (which is defined in terms of the neck) comes along for the ride.
So how do we get this relative coordinate system?  With a change of basis!  A change of basis is really just saying, "Pretend we're starting in the right place without any transformations applied.  Then apply the transformations when we're done."