When I ask you to picture the vector , most of you see this in your head:
To be precise, what you are picturing is the vector in the standard basis for : , where represents a unit vector in the direction of the positive -axis, and represents a unit vector in the direction of the positive -axis.
We write , and we think of as being the coordinates of this vector with respect to this basis.
Why this pedantry? We might, for example, want to express the vector above as a linear combination of different basis vectors, e.g. :
Using the notation above, we have
The vector has coordinates in this new basis.
Is there a way to get directly from the LHS to the RHS? The theorem below tells us how to:
Let and be two bases for a given space. Let be an matrix such that for any , . (The th column of is expressed as a linear combination in .)
Then, for any , .
The proof consists solely of matrix algebra:
Let’s see this theorem in action for our example above. With respect to the standard coordinate system, we have
which means . Hence,
as we saw earlier.