A Note on Course Practice and the Course Project
Published:
Last Update: 2026-03-22
(Chinese version follows the English text / 中文版见后)
Preface
Starting from the Spring 2025 semester, traditional homework assignments will no longer be factored into the final grade for this course. Instead, the continuous assessment will be replaced by a Course Project, accounting for 25% of the total score.
This adjustment is part of a unified administrative mandate, the underlying motivations of which are self-evident. However, it is worth noting that such a “reform” deviates significantly from the fundamental learning patterns required for foundational mathematics. The efficacy of similar arrangements in practical teaching is well-documented by historical precedents; as for its long-term validity and actual impact, we shall let the subsequent academic outcomes provide the answer.
Regarding the course itself, I wish to emphasize one fundamental fact that remains independent of any grading system:
The mastery of foundational mathematics relies on consistent practice—a process that cannot be substituted by any form of pedagogical “design.”
Linear Algebra is not inherently difficult, yet its mode of thinking differs substantially from high school mathematics or even Calculus. One must engage in sufficient practice to master core computations (such as matrix row reduction and elimination, which form the bedrock of nearly all calculations in this course). Only upon this foundation does a genuine understanding of definitions, concepts, and theorems become possible.
Attempting to compensate for a semester’s lack of practice with intensive “cramming” during the finals is, in the vast majority of cases, futile. This is not merely a personal observation but a repeatedly verified conclusion. Therefore, let it be clear: the removal of homework from the grading scheme does not lower the requirement for practice; it only lowers the incentive for some to begin. This discrepancy will manifest directly and unmistakably during the final examination.
Consequently, even if homework is no longer graded, I strongly recommend completing assignments according to the course schedule and supplementing them with additional exercises as needed. This is not an “extra” demand, but a basic prerequisite for successfully completing this course.
Course Project
Regarding the Project itself, the original intent is to integrate linear algebra methods with engineering problems, moving beyond rote examination techniques. Conceptually, this is sound: it facilitates training in mathematical modeling, academic writing, teamwork, and presentation skills. As a practical component, it offers more value than most purely exam-oriented curricula.
However, it must be underscored that these positive outcomes depend on one crucial premise: every team member must genuinely master the relevant knowledge and conscientiously fulfill their assigned role.
If you are committed to completing this task rigorously or hope to gain authentic insights from it, here are my suggestions:
Given the finite set of tools available in this course (linear systems, matrix operations, determinants, linear transformations, eigenvalues/eigenvectors, and basic orthogonality/quadratic forms), your topic selection should prioritize problems you can independently solve and clearly articulate, rather than pursuing “sophisticated” or overly complex tasks.
Feasible directions include, but are not limited to:
Algorithmic Implementation: Coding core course algorithms (e.g., Row Echelon Form, Gaussian Elimination, 2D/3D Linear Transformations).
Focused Extensions: Exploring practical engineering methods not covered in depth during lectures, such as SVD, QR Decomposition, or Least-Squares Fitting.
You are expected to produce a standard academic report in English describing a real-world engineering problem you have addressed. The evaluation will not focus on length or the stacking of abstract concepts, but rather on:
- Clarity of the problem definition.
- Effective application of linear algebra tools and methods.
- Transparency and rigor of the computational process.
- Rationality and self-consistency of the conclusions.
The overall logic must be complete: Problem $\rightarrow$ Model $\rightarrow$ Computation $\rightarrow$ Result. For group collaboration, please ensure a reasonable division of labor and individual accountability (the workflow of Mathematical Contest in Modeling (MCM) may serve as a useful reference).
（中文版）
写在前面的话
从 2025 年春季学期起，本课程不再将传统平时作业计入总成绩，过程考核由占比 25% 的课程项目（Project）替代。
这一调整属于统一教学安排，其背后的动因不难理解。这一 “改革” 与数学类基础课程的基本学习规律明显背离，类似安排在实际教学中的效果如何，过往并不缺乏可参考的例子。至于其合理性与实际效果，不妨交由后续教学结果来回答。
就课程本身而言，我只强调一个与任何考核制度无关的基本事实：
数学类基础课程的学习依赖持续练习，这一点无法被任何形式的 “设计” 所替代。
线性代数本身难度并不算高，但其思维方式与中学数学乃至微积分均有显著差异。必须通过足量练习，才能熟练掌握核心计算（如矩阵行化简、消元——这些构成了课程几乎所有计算的基础）；也只有在此基础上，才可能真正理解定义、概念与定理。
整学期不进行练习，而寄希望于期末阶段集中补救，在绝大多数情况下不可行 —— 这不是个体经验，而是反复被验证的结果。
因此需要明确一点：取消作业成绩，并不会降低这门课对练习的要求，只会降低一部分人开始练习的动力。这两者之间的差别会在期末阶段体现得非常直接。
因此，即便平时作业不再计入总成绩，仍建议按照课程进度完成作业，并根据自身情况补充必要练习 —— 这不是额外要求，而是完成本课程学习的基本条件。
课程项目（Project）
单纯就 Project 本身而言，其初衷是将线性代数方法与工程问题相结合，而非停留在应试技巧。从设计理念上看，这是合理的：它可以训练建模能力、学术写作、团队协作与展示能力，作为实践环节，比大多数纯应试型课程更有价值。
但必须强调：这一切积极效果，前提只有一个 —— 团队中每位成员都真正掌握相关知识并认真完成分工。
如果你愿意认真完成，或是想从中获得一点真实收获，以下是我的一些个人建议：
在本课程可使用工具有限的前提下（线性方程组、矩阵运算、行列式、线性变换、特征值与特征向量、基础正交 / 二次型），选题应遵循：选择自己能够完成、能够讲清楚的问题，而非追求 “高大上” 或复杂任务。
可行方向包括但不限于：
- 动手实现课程算法（矩阵行化简、高斯消元、二维 / 三维线性变换等）
- 在课程基础上做有限延伸，了解一些工程中常用但课上未深入讲解的方法，如 SVD、QR 分解、最小二乘拟合等
请完成一次规范的英文学术写作，描述真实动手做过的工程问题。报告不看篇幅、不看概念堆砌，只看：
- 问题是否明确
- 是否真正使用了线性代数工具与方法
- 计算过程是否清晰
- 结论是否合理自洽
整体逻辑必须完整：问题 → 模型 → 计算 → 结果。
小组合作请合理分工、各负其责（可参考数学建模竞赛模式）。