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.

Given that the two vector spaces start from the origin of coordinates in the XYZ space...I believe I can eliminate translation and simply solve for a 3X3 transformation matrix...is this accurate? Both spaces are contained within the gamut of color visible to humans...this would place constraints on all color spaces..thus the conclusion I can rule out translations.