10.求向量组alpha_(1)=(1,0,2,1)^T,alpha_(2)=(1,2,0,1)^T,alpha_(3)=(2,1,3,0)^T,alpha_(4)=(2,5,-1,4)^T,alpha_(5)=(1,-1,3,-1)^T的秩及一个极大无关组、并将其余向量用该极大无关组线性表示.

将向量组构成矩阵 $A$，进行初等行变换化为行最简形： \[ A = \begin{pmatrix} 1 & 1 & 2 & 2 & 1 \\ 0 & 2 & 1 & 5 & -1 \\ 2 & 0 & 3 & -1 & 3 \\ 1 & 1 & 0 & 4 & -1 \end{pmatrix} \rightarrow \begin{pmatrix} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 3 & -1 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} \] **秩**：3 **极大无关组**：$\alpha_1, \alpha_2, \alpha_3$ **线性表示**： \[ \alpha_4 = \alpha_1 + 3\alpha_2 - \alpha_3, \quad \alpha_5 = -\alpha_2 + \alpha_3 \] \[ \boxed{ \begin{array}{ccc} \text{秩：} & 3 \\ \text{极大无关组：} & \alpha_1, \alpha_2, \alpha_3 \\ \text{线性表示：} & \alpha_4 = \alpha_1 + 3\alpha_2 - \alpha_3, \quad \alpha_5 = -\alpha_2 + \alpha_3 \end{array} } \]