Galoitica: Journal of Mathematical Structures and Applications GJMSA 2834-5568 10.54216/GJSMA https://www.americaspg.com/journals/show/2057 2022 2022 University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China Lee Xu This paper is dedicated to find Legendre polynomials by using novel linear algebraic methods based on matrices based on Liouville-Sturm theorem, where we find the matrix of the differential operator for Legendre polynomials, with their eigenvalues and their eigenvectors. Also, we illustrate many examples to clarify the validity of our work. 2023 2023 47 50 10.54216/GJMSA.070205 https://www.americaspg.com/articleinfo/33/show/2057