In most textbooks on mechanics, electrodynamics and relativity, the linear transformation between inertial frames is usually argued by the homogeneity and isotropy of spacetime. During teaching this subject, some students cannot understand this simple argument. After all, the science of physics should be explained with solid mathematics. This paper provides a solid mathematical proof of linear transformation between inertial frames based on the two postulates of special relativity and the homogeneity and isotropy of spacetime.