基、线性方程与秩Bases, Linear Systems & Rank

从列的组合出发Start with column combinations

Ac=b 意味着用 A 的列向量乘以系数后相加，得到目标 b。两根不共线的二维向量能表示平面中任意目标；它们构成该平面的一组基。Ac=b means multiplying the columns of A by coefficients and adding them to reach b. Two non-collinear vectors in two dimensions represent any target in the plane; they form a basis of that plane.

退化会丢失方向Degeneracy loses a direction

若两列共线且至少一列非零，只能到达一条直线，秩为 1。目标在线上时通常有多组系数；目标在线外时无解。零矩阵秩为 0，只能到达原点。If the columns are collinear and at least one is nonzero, they reach only a line and have rank 1. A target on that line typically has multiple coefficient solutions; one off the line has none. The zero matrix has rank 0 and reaches only the origin.

解与秩Solutions and rank

二维方阵的两列线性无关时，任意目标有唯一解。更一般的非方阵中，满列秩只能保证“若有解则唯一”，不能保证所有目标都有解。For a two-dimensional square matrix with independent columns, every target has a unique solution. For a general rectangular matrix, full column rank guarantees only that a solution, if one exists, is unique; not every target must be reachable.

公式与符号Formula and symbols

A 的列是可组合的向量，c 是组合系数，b 是目标；rank(A) 是线性无关的列方向数。线性无关表示只有所有系数为零时才能组合出零向量。The columns of A are vectors to combine, c contains coefficients, and b is the target. rank(A) is the number of independent column directions. Independent means only all-zero coefficients combine to the zero vector.

继续探索Explore further

• 深入阅读：矩阵变换与行列式Further reading: matrix transformations and determinants