定理1

向量组 \(\mathbf{\alpha}_{1},\mathbf{\alpha}_{2},\mathbf{\alpha}_{3},\dots,\mathbf{\alpha}_{n} (n\geq 2)\) 线性相关的充要条件是向量组中至少有一个向量可由其余的 \(n-1\) 个向量线性表示.

定理2

若向量组 \(\mathbf{\alpha}_{1},\mathbf{\alpha}_{2},\mathbf{\alpha}_{3},\dots,\mathbf{\alpha}_{n} (n\geq 2)\) 线性无关，而 \(\mathbf{\beta},\mathbf{\alpha}_{1},\mathbf{\alpha}_{2},\mathbf{\alpha}_{3},\dots,\mathbf{\alpha}_{n}\) 线性相关，则 \(\mathbf{\beta}\) 可由 \(\mathbf{\alpha}_{1},\mathbf{\alpha}_{2},\mathbf{\alpha}_{3},\dots,\mathbf{\alpha}_{n}\) 线性表示，且表示法唯一.

定理3

若向量组 \(\mathbf{\beta}_{1},\mathbf{\beta}_{2},\mathbf{\beta}_{3},\dots,\mathbf{\beta}_{t}\) 可由向量组 \(\mathbf{\alpha}_{1},\mathbf{\alpha}_{2},\mathbf{\alpha}_{3},\dots,\mathbf{\alpha}_{s}\) 线性表示，且 \(t>s\) 则 \(\mathbf{\beta}_{1},\mathbf{\beta}_{2},\mathbf{\beta}_{3},\dots,\mathbf{\beta}_{t}\) 线性相关

口诀：以少表多，多的相关

定理4

设 \(m\) 个 \(n\) 维向量 \(\mathbf{\alpha}_{1},\mathbf{\alpha}_{2},\mathbf{\alpha}_{3},\dots,\mathbf{\alpha}_{m}\) ，其中

\[ \mathbf{\alpha _{1}}=\left[ \alpha _{11},\alpha _{21},\ldots ,\alpha _{n1}\right] ^{T}, \] \[ \mathbf{\alpha _{2}}=\left[ \alpha _{12},\alpha _{22},\ldots ,\alpha _{n2}\right] ^{T}, \] \[ \dots\dots \] \[ \mathbf{\alpha _{m}}=\left[ \alpha _{1m},\alpha _{2m},\ldots ,\alpha _{nm}\right] ^{T}. \] 向量组 \(\mathbf{\alpha}_{1},\mathbf{\alpha}_{2},\mathbf{\alpha}_{3},\dots,\mathbf{\alpha}_{m}\) 线性相关$\Leftrightarrow $ 齐次线性方程组

\[ \left[ \alpha _{1},\alpha _{2},\ldots \alpha _{m}\right] \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{m} \end{bmatrix}=x_{1}\alpha _{1}+x_{2}\alpha _{2}+\ldots +x_{m}\alpha _{m}=\mathbf{0} \]

有非零解 \(\Leftrightarrow r(\alpha _{1},\alpha _{2},\dots,\alpha _{m})<m .\)

其等价命题：\(\mathbf{\alpha}_{1},\mathbf{\alpha}_{2},\mathbf{\alpha}_{3},\dots,\mathbf{\alpha}_{m}\) 线性无关的充分必要条件是齐次线性方程组 \((*)\) 只有零解.