Abstract: The linear transformation properties on the jointly normally distributed of random vector are very important in modeling analysis in many problems.This paper has proved by matrix analysis,that the new random vector based on linear transform of n dimension joint normal random vector,has a joint normal distribution only if the linear transformation is with row full rank.Otherwise,it is subject to the degenerate joint normal distribution.In this paper,a numerical experiment is designed to simulate and analyze the theoretical results.The experimental results show the randomness of the points under different transformations.