设 $\alpha _1, \alpha _2, \alpha _3$ 是 3 维向量空间 $R ^3$ 的一组基, 则由基 $\alpha _1, \frac{1}{2} \alpha _2, \frac{1}{3} \alpha _3$ 到基 $\alpha _1+ \alpha _2, \alpha _2+ \alpha _3, \alpha _3+ \alpha _1$ 的过渡矩阵为
$\text{A.}$ $\left(\begin{array}{lll}1 & 0 & 1 \\ 2 & 2 & 0 \\ 0 & 3 & 3\end{array}\right)$.
$\text{B.}$ $\left(\begin{array}{lll}1 & 2 & 0 \\ 0 & 2 & 3 \\ 1 & 0 & 3\end{array}\right)$.
$\text{C.}$ $\left(\begin{array}{ccc}\frac{1}{2} & \frac{1}{4} & -\frac{1}{6} \\ -\frac{1}{2} & \frac{1}{4} & \frac{1}{6} \\ \frac{1}{2} & -\frac{1}{4} & \frac{1}{6}\end{array}\right)$.
$\text{D.}$ $\left(\begin{array}{ccc}\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\ \frac{1}{4} & \frac{1}{4} & -\frac{1}{4} \\ -\frac{1}{6} & \frac{1}{6} & \frac{1}{6}\end{array}\right)$.