📚 Analysis of Nearly Linear Systems | 近线性系统的分析
In IB Mathematics, a linear system is a collection of linear equations involving the same set of variables. The analysis of such systems forms a core part of linear algebra, and it extends naturally to ‘nearly linear’ systems where small nonlinearities are approximated using linear techniques. This article explores the key ideas, tools, and exam-relevant methods for analysing these systems.
在IB数学中，线性系统是由同一组变量构成的多个线性方程的集合。分析这类系统是线性代数的重要部分，并自然延展到“近线性”系统——即借助线性化技术来近似处理小规模非线性。本文将探讨分析这些系统的关键思想、工具及与考试相关的方法。
1. Linear Systems and Their Forms | 线性系统及其形式
A linear system of equations can be written in the general form:
线性方程组的一般形式可以写成：
a₁₁x₁ + a₁₂x₂ + … + a₁ₙxₙ = b₁
a₂₁x₁ + a₂₂x₂ + … + a₂ₙxₙ = b₂
⋮
aₘ₁x₁ + aₘ₂x₂ + … + aₘₙxₙ = bₘ
Here aᵢⱼ are coefficients, xⱼ are unknown variables, and bᵢ are constants. If all bᵢ = 0, the system is homogeneous; otherwise it is non-homogeneous.
其中aᵢⱼ为系数，xⱼ为未知变量，bᵢ为常数。若所有bᵢ = 0，则称为齐次系统；否则为非齐次系统。
A single linear equation represents a plane in 3D or a hyperplane in higher dimensions. The solution set of a linear system is the intersection of these geometric objects.
一个线性方程在三维空间中表示一个平面，在更高维表示超平面。线性方程组的解集就是这些几何对象的交集。
2. Matrix Representation | 矩阵表示
Using matrix notation, the system becomes Ax = b, where A is the coefficient matrix, x is the variable column vector, and b is the constant vector. For example:
利用矩阵记号，方程组可写成Ax = b，其中A是系数矩阵，x是未知数列向量，b是常数向量。例如：
A = [aᵢⱼ], x = [x₁ x₂ … xₙ]ᵀ, b = [b₁ b₂ … bₘ]ᵀ
The augmented matrix [A | b] is often used for row-reduction methods such as Gaussian elimination. In IB, you should be comfortable transforming between the equation form and the matrix form.
增广矩阵[A | b]常用于高斯消元法等行化简方法。在IB考试中，你需要能够熟练地在方程组形式与矩阵形式之间转换。
3. Solution Structure: Unique, None, Infinite | 解的结构：唯一解、无解、无穷多解
For a square system (number of equations = number of unknowns), the determinant of A plays a key role.
对于方阵系统（方程数等于未知数数），A的行列式起着关键作用。
- If det(A) ≠ 0, the system has a unique solution given by x = A⁻¹b.
- If det(A) = 0, the system either has no solution or infinitely many solutions.
- 若det(A) ≠ 0，方程组有唯一解，且x = A⁻¹b。
- 若det(A) = 0，方程组要么无解，要么有无穷多解。
To distinguish between ‘no solution’ and ‘infinite solutions’, we use the rank of the coefficient matrix and the augmented matrix. If rank(A) < rank([A | b]), the system is inconsistent.
为区分“无解”和“无穷多解”，需比较系数矩阵与增广矩阵的秩。若rank(A) < rank([A | b])，则方程组无解；若二者相等且小于未知量个数，则有无穷多解。
4. Eigenvalues and Eigenvectors | 特征值与特征向量
An eigenvector v of a matrix A satisfies Av = λv, where λ is the corresponding eigenvalue. These are found by solving the characteristic equation:
矩阵A的特征向量v满足Av = λv，其中λ是对应的特征值。通过特征方程求解：
det(A − λI) = 0
Eigenvalues give deep insight into the behaviour of a linear system. In IB Mathematics, you may be asked to find eigenvalues and eigenvectors of 2×2 or 3×3 matrices, and to use them to diagonalise a matrix or solve systems of differential equations.
特征值能够深入揭示线性系统的行为。在IB数学中，你可能需要求2×2或3×3矩阵的特征值与特征向量，并利用它们对角化矩阵或求解微分方程组。
5. Linear Systems of Differential Equations | 线性微分方程组
Consider a linear system of first-order differential equations:
考虑一阶线性微分方程组：
dx/dt = Ax
where x is a vector of functions of t. If A is diagonalisable, the solution can be written using eigenvalues and eigenvectors:
其中x是t的函数向量。若A可对角化，则解可以用特征值和特征向量表示：
x(t) = c₁e^(λ₁t)v₁ + c₂e^(λ₂t)v₂ + …
The real parts of the eigenvalues determine whether solutions grow, decay, or oscillate. This leads directly to the concept of stability.
特征值的实部决定了解是增长、衰减还是振荡。这直接引向稳定性的概念。
6. Nearly Linear Systems and Linearisation | 近线性系统与线性化
A ‘nearly linear’ system is a nonlinear system whose behaviour near a specific point is close to that of a linear system. The key tool is the Jacobian matrix.
“近线性”系统是指非线性系统在某个特定点附近的行为非常接近线性系统。核心工具是雅可比矩阵。
Given a nonlinear system dx/dt = f(x), the equilibrium points satisfy f(x*) = 0. To analyse behaviour near x*, we compute the Jacobian matrix J(x*), whose entries are partial derivatives:
给定非线性系统dx/dt = f(x)，平衡点满足f(x*) = 0。为分析x*附近的行为，我们计算雅可比矩阵J(x*)，其元素是偏导数：
Jᵢⱼ = ∂fᵢ / ∂xⱼ
Then the linearised system near x* is du/dt = J(x*)u, where u = x − x*. This approximation is valid for small perturbations from equilibrium.
于是x*附近的线性化系统为du/dt = J(x*)u，其中u = x − x*。该近似在平衡点附近的小扰动下有效。
7. Stability Analysis at Equilibrium Points | 平衡点的稳定性分析
The stability of an equilibrium point is determined by the eigenvalues of the Jacobian matrix at that point.
平衡点的稳定性由该点处雅可比矩阵的特征值决定。
| Eigenvalues λ | Type / Stability | 类型 / 稳定性 |
| All Re(λ) < 0 | Stable node / spiral (asymptotically stable) | 稳定结点/螺旋（渐近稳定） |
| All Re(λ) > 0 | Unstable node / spiral | 不稳定结点/螺旋 |
| Mixed signs of Re(λ) | Saddle point (unstable) | 鞍点（不稳定） |
| Re(λ) = 0 | Centre or degenerate case | 中心或退化情形 |
This powerful idea connects linear algebra to dynamical systems, and appears in IB Mathematics as an extension topic or in the context of modelling.
这个强有力的思想将线性代数与动力系统联系起来，并作为拓展主题或在建模情境中出现在IB数学中。
8. Real-World Applications and Modelling | 实际应用与建模
Nearly linear systems appear in physics (pendulum motion, circuits), biology (predator-prey models), economics (market equilibrium) and engineering (control systems).
近线性系统出现在物理学（单摆运动、电路）、生物学（捕食者-猎物模型）、经济学（市场均衡）和工程学（控制系统）等领域。
For example, the simple pendulum equation θ” + (g/L)sinθ = 0 is nonlinear. For small θ, sinθ ≈ θ, so the system becomes nearly linear:
例如，单摆方程θ” + (g/L)sinθ = 0是非线性的。当θ很小时，sinθ ≈ θ，因此系统变为近线性：
θ” + (g/L)θ = 0
The linearised solution then provides a very good approximation to the true motion, and the error is of order θ³.
线性化解在此时能很好地逼近真实运动，误差为θ³的量级。
9. Summary and IB Exam Tips | 总结与IB考试提示
To master the analysis of nearly linear systems, remember the following steps:
要掌握近线性系统的分析，请记住以下步骤：
- Write the system in matrix form when possible.
- Use row reduction or determinants to classify solutions.
- Find eigenvalues and eigenvectors to understand long-term behaviour.
- For nonlinear systems, identify equilibrium points and linearise using the Jacobian.
- Link the signs of real parts of eigenvalues to stability.
- 尽可能将系统写成矩阵形式。
- 使用行化简或行列式来分类解。
- 求特征值和特征向量，以理解长期行为。
- 对于非线性系统，找到平衡点并用雅可比矩阵进行线性化。
- 将特征值实部的符号与稳定性联系起来。
In IB exams, clearly state your method, show the augmented matrix, and justify why a system is ‘nearly linear’ before applying linear analysis. Practice with past papers to become fluent in these techniques.
在IB考试中，请清晰说明你的方法，写出增广矩阵，并在应用线性分析之前论证为何系统是“近线性”的。通过练习历年真题，熟练掌握这些技巧。
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