We study the relations among various linear mapping-based algorithms by formulating a more general unified pseudo-inverse algorithm. We show that the least-square linear mapping technique, the simplified least-square linear mapping technique, the synthetic discriminant function, the equal correlation peak method and the Caulfield-Maloney filter are in fact all special cases of the unified pseudo-inverse algorithm. When the total number of the training images (KM, where K is the number of classes and M is the number of training images in each class) is larger than the dimension of the images (N), the overdetermined case of the unified pseudo-inverse algorithm is the same as the least-square linear mapping technique, due to the fact that both algorithms are based on optimization processes of minimization of the least square error. When KM < N, the underdetermined case of the unified pseudo-inverse algorithm is the same as the least-square linear mapping technique and the synthetic discriminant function. Furthermore, when KM < N, the synthetic discriminant function method can be considered as the degenerated case of the least-square linear mapping technique. Experimental results on classification using the linear mapping-based algorithms are provided and show good agreement with the theoretical analysis.