Matching pure primaries individually is the procedure for deriving a color space conversion matrix

To find the 3×3 matrix that converts linear color space A to linear color space B, match pure red, pure green, and pure blue from A to their corresponding values in B separately. Each match produces one column of coefficients (how much R, G, B each primary contributes). For example, matching pure R from A into B gives three numbers that become the first column. Then Grassmann’s law allows these three individual matches to be combined into one set of equations for any mixed color. This is how ICC color profiles are derived.

Examples

Matching pure R=1.0 from wide-gamut space into narrower space gives: R’=0.712, G’=0.100, B’=0.024 — these are the first column of the forward conversion matrix.

Assessment

Describe the steps for deriving the conversion matrix from color space A to B. Why does this need to be done with linear (not gamma-encoded) values?