Section 3.4 Change of Bases/Coordinates
¶Subsection 3.4.1 Coordinate Systems
permalinkSuppose that we want to walk to the point (2,3) from the origin, how might we get there?
We can get from the origin to the point (2,3) by walking east 2 blocks and north 3.

We can get there by walking northeast(ish) 3 blocks and northwest(ish) 1.

We can get there by walking northwest(ish) 2 blocks and northeast(ish) 4.

permalinkIn each case you get to the same place but using different paths. That is because in each case we are using different bases or coordinate systems.
permalinkIn Figure 3.4.1 we follow the vectors in the basis
permalinkWe say that the E2-coordinates for (2,3) are
permalinkbecause
permalinkIn Figure 3.4.2 we follow the vectors in the basis
permalinkWe say that the B-coordinates for (2,3) are
permalinkbecause
permalinkIn Figure 3.4.3 we follow the vectors in the basis
permalinkWe say that the D-coordinates for (2,3) are
permalinkbecause
permalinkIn each case the coordinates for the point (2,3) are the coefficients for a linear combination of basis vectors.
Definition 3.4.4. B-Coordinates.
Given a basis
the B-coordinates for a point or vector →p are the coefficients b1,b2,…,bk so that
and we write
permalinkFinally, we can connect this to matrices by observing that if the columns of a matrix are the basis vectors then when we multiply that by a representation we get our point, for example
Subsection 3.4.2 Changing Bases
Investigation 3.4.1. B to E2.
The elementary or standard basis is
Let B be the basis
as above. If we have a vector written in terms of B,
and we want RepE2(→v) then we multiply by the matrix
So we get,
To change from any basis
to Ek multiply by the matrix
each of whose columns is an element of B.
Investigation 3.4.2. E2 to B.
Let E2 and B be the same as before and suppose
to change from E2 to B multiply by the matrix
So we get,
To change from Ek to any basis
multiply by the matrix
Investigation 3.4.3. B to D.
Finally, let
and let B and E2 be as before. Suppose also that
if we want to find RepD(→v) then multiply by the matrix
So that we get
We can verify that this is correct because
and
In general if
and
then
Investigation 3.4.4. Back to the Beginning.
We started above with
Now, we know that if we multiply the first one,
by
or
we get the other two. That is we have a reliable, algorithmic way to change from one basis to another in the same dimension.
permalinkSummary Diagram for Bases Changes:

Subsection 3.4.3 Changing Morphisms
Investigation 3.4.5. Change a transformation of E2 to one from B to D.
Let E2 be as bove and let
Define a transformation P from R3 to R2 by
From before the matrix for P is
and it maps E3 to E2.
Now suppose that we want the transformation to change vectors with coordinates in the basis
to vectors in terms of the basis
Do this by going from B to E3 to E2 to D,
Using this we get
permalinkSummary Diagram for Transformation Changes:
