面向新工科人才培养的线性代数教学改革与实践——以“计算机图形学”为载体的线性空间与线性变换创新教学设计

Reform and Practice of Linear Algebra Teaching for Cultivating New Engineering Talents—An Innovative Instructional Design of Linear Spaces and Linear Transformations Using “Computer Graphics” as the Carrier

摘要: 由于线性代数课程中线性空间及线性变换章节抽象性极强，学生理解困难，又存在学用分离的突出教学问题，因此本文提出了以“计算机图形学”为真实应用背景的项目式教学改革方案，把抽象的向量空间、线性变换、矩阵表示、特征值诸概念系统、有层次地映射到二维图像处理及三维模型视图变换的具体任务中，由此构造出“现象感知–数学建模–编程实现–应用拓展”的四阶学习闭环。更难得的是，借助“图像旋转缩放”和“三维模型视图变换”两个递进式实践项目，真正引导学生从被动接受转向主动探究，在解决实际工程问题的过程中自发、充分地掌握数学原理。教学实践结果十分清楚地证实，该模式有利于提高学生的几何直观能力、计算思维能力及知识迁移能力，也因而激发了学习兴趣，故而成为新工科背景下基础数学课程教学改革的可复制、可推广的范式。

Abstract: Due to the high level of abstraction in the chapters on linear spaces and linear transformations within the linear algebra curriculum, students often find them difficult to understand, and a prominent issue of separation between learning and application exists. To address this, this paper proposes a project-based teaching reform scheme using “Computer Graphics” as a real-world application context. It systematically and hierarchically maps abstract concepts such as vector spaces, linear transformations, matrix representations, and eigenvalues to specific tasks involving 2D image processing and 3D model view transformation. This approach constructs a “phenomenon perception-mathematical modeling-programming implementation-application expansion” four-phase learning cycle. Notably, through two progressive practical projects—“image rotation and scaling” and “3D model view transformation”—students are genuinely guided from passive reception to active inquiry, mastering mathematical principles spontaneously and thoroughly while solving practical engineering problems. Teaching practice results clearly demonstrate that this model helps enhance students’ geometric intuition, computational thinking, and knowledge transfer abilities, thereby stimulating their learning interest. Consequently, it serves as a replicable and scalable paradigm for reforming fundamental mathematics courses within the context of New Engineering education.