Abstract: Integrating ideological and political education into courses is a strategic measure to achieve the fundamental task of fostering virtue through education. Addressing the imperative to advance the ideological and political construction within the linear algebra course, the core essence of mathematical culture and ideological and political education in courses are analyzed. Based on a survey of 262 teachers and students, it examines the issues in the teaching of linear algebra courses. Leveraging the highly compatible characteristics of the immersive educational nature of mathematical culture and the implicit educational principles of ideological and political education, a "5E + Culture" integrated teaching model for ideological and political education in courses is constructed. The teaching of "inverse matrices" is used as a case study for empirical validation,evaluating and analyzing the teaching effectiveness under this model are also carried out. Research shows that this model naturally incorporates cultural elements, national sentiment, and scientific spirit during the knowledge construction process. The evaluation phase emphasizes self-assessment by both teachers and students, thereby promoting continuous improvement in teaching quality. This provides a theoretical model and practical paradigm for the development of ideological and political education in linear algebra courses.