Rank

rank(A) = the dimension of column space of A, i.e., \(\dim\text{span }\{a_1, \cdots, a_n\}\)

• \(\text{rank} (A+B) \leq \text{rank}(A) + \text{rank}(B)\).

秩的各种定义

行秩列秩相等性

基本证明思路：

列秩为像空间的维度，行秩为非零原像空间的维度

• \(\text{column rank} = \dim \Im(A) = \dim \{Ax: x\in \mathbb{R}^{n}\}\).
• \(\text{row rank} = \dim \{x\in\mathbb{R}^{n} : Ax\neq 0\}\).

证明一

证明二

证明三

秩-零化度定理(Rank-nullity theorem)