A derivation of the procedure for changing bases (for )

Suppose we have Source Basis S consisting of vectors: s1 and s2
And Target Basis T consisting of vectors: t1 and t2
And a vector v expressed in S as: = {x,y}

Each source base vector can be expressed as a linear combination of the target base vectors. That is:

s1 = k1 t1 + k2 t2
s2 = c1 t1 + c2 t2

Note that s1 expressed in T is: {k1, k2} and s2 expressed in T is: {c1, c2}

If we express v as a linear combination of the source vectors, we can see that:

v = x s1 + y s2
    = x k1 t1 + x k2 t2 + y c1 t1 + y c2 t2
    = (x k1 + y c1)t1 + (x k2 + y c2)t2

But that means v is expressed as:
     {x k1 + y c1,  x k2 + y c2}
In the target basis.

The same transformation can be achieved by constructing a matrix with the  the source vectors expressed in the target basis as the columns and using it to mutiply the vetor of interest:

This shows that a vector can be transformed from one basis to another by multiplying it by the matrix whose columns are the source basis vectors expressed in the target basis.

Created by Mathematica  (August 5, 2007)