3D affine transformation question

I have an application that I am building that requires an affine transformation between two vector spaces. I am using the CIE 1931 xyY

CIE 1931 color space - Wikipedia, the free encyclopedia

color space as the starting point for two color spaces. I transform both color spaces to XYZ space (a true vector space). At this point, I would like to apply an affine transformation to map one space onto the other.

My question is, when I get to the point where I am solving the affine transformation matrix (a 4X4) matrix, how can I constrain the equations given only the three points of the vector space? Do I simply use a 4th point?

The matrix eqations take the form of the image below:

http://cse.taylor.edu/~btoll/s99/424.../3d-affine.jpg

Just to be clear...I know the vectors...I need the values in the matrix.