6.2 基$\cdot$维数与坐标 - 线上练习
满分: 28分
  1. (填空题, 每空2分) 在线性空间$P^{2×2}$中, 考察矩阵 \[A_1 = \begin{pmatrix} a & 1 \\ 1 & 1 \end{pmatrix}, A_2 = \begin{pmatrix} 1 & a \\ 1 & 1 \end{pmatrix}, A_3 = \begin{pmatrix} 1 & 1 \\ a & 1 \end{pmatrix}, A_4 = \begin{pmatrix} 1 & 1 \\ 1 & a \end{pmatrix},\] 则当$a=\underline{\qquad},\underline{\qquad}$时, $A_1,A_2,A_3,A_4$线性相关.
  2. (单选题, 4分) 设$V$是数域$P$上的线性空间, 考察命题:
    $\RI$ 若$\alpha_1,\alpha_2,\alpha_3\in V$线性无关, 则$\alpha_1+\alpha_2,\alpha_2+\alpha_3,\alpha_3+\alpha_1$也线性无关,
    $\RII$ 若$\alpha_1,\alpha_2,\alpha_3,\alpha_4\in V$线性无关, 则$\alpha_1+\alpha_2,\alpha_2+\alpha_3,\alpha_3+\alpha_4,\alpha_4+\alpha_1$也线性无关,
    则$(\qquad)$.




  3. (填空题, 每空4分) 设$V$是数域$P$上的线性空间, 记$V$中向量组: \[\RI\, \alpha_1,\alpha_2,\alpha_3, \quad \RII\, \alpha_1,\alpha_2,\alpha_3,\alpha_4, \quad \RIII\, \alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5.\] 若这三个向量组的秩分别为$R\RI=R\RII=3, R\RIII=4$, 则向量组$\alpha_1,\alpha_2,\alpha_3,\alpha_5−\alpha_4$的秩为 $\underline{\qquad}$. (提示: 考察$\alpha_5−\alpha_4$是否可由$\alpha_1,\alpha_2,\alpha_3$线性表出.)
  4. (单选题, 4分) 设$V$是数域$P$上的线性空间, $\alpha_1,\alpha_2,\alpha_3\in V$线性无关, 则下列向量组中与$\alpha_1,\alpha_2,\alpha_3$等价是$(\qquad)$. (提示: 考察各选项中向量组的秩.)




  5. (单选题, 4分) 设$V$是数域$P$上的线性空间, $\alpha_1,\alpha_2,\alpha_3,\alpha_4$是$V$的一组基, 以下哪个向量组也是$V$的基?




  6. (填空题, 每空4分) 设$P$是数域, 在线性空间$P^4$中, 设向量组 \[\alpha_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 3 \end{pmatrix}, \quad \alpha_2 = \begin{pmatrix} 1 \\ 1 \\ -1 \\ 2 \end{pmatrix}, \quad \alpha_3 = \begin{pmatrix} 1 \\ 2 \\ a-3 \\ 1 \end{pmatrix}, \quad \alpha_4 = \begin{pmatrix} 1 \\ 2 \\ -2 \\ a \end{pmatrix},\] 则$\alpha_1,\alpha_2,\alpha_3,\alpha_4$是$P^4$的基当且仅当$a\neq\underline{\qquad}$.
  7. (填空题, 每空1分) 设$P$是数域,在线性空间$P^{2×2}$中, 取一组基 \[A_1 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad A_2 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad A_3 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad A_4 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},\] 则$B = \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix}$ 在这组基下的坐标的第1个分量为 $\underline{\qquad}$, 第2个分量为 $\underline{\qquad}$, 第3个分量为$\underline{\qquad}$, 第4个分量为$\underline{\qquad}$.
注: 填空题如有多个无序答案, 按从小到大顺序填写.