Example. The vector $\mathbf{x}$ in the basis $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ has coordinates $\{1,2,3\}$. Find the coordinates of the vector $\mathbf{x}$ in the basis $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime}$, where

$$
\begin{aligned}
& \mathbf{e}_{1}^{\prime}=\mathbf{e}_{1}+2 \mathbf{e}_{3} \\
& \mathbf{e}_{2}^{\prime}=\mathbf{e}_{2}+\mathbf{e}_{3} \\
& \mathbf{e}_{3}^{\prime}=-\mathbf{e}_{1}-\mathbf{e}_{2}-2 \mathbf{e}_{3}
\end{aligned}
$$
Solution.

1. Find the transition matrix

$$
C=\left(\begin{array}{ccc}
1 & 0 & -1 \\
0 & 1 & -1 \\
2 & 1 & -2
\end{array}\right)
$$

2. Find the inverse matrix $C^{-1}$ using the Gaussian method:

![](https://cdn.mathpix.com/cropped/2024_05_22_5878ce417e2ca061000bg-064.jpg?height=153&width=965&top_left_y=1126&top_left_x=144)

Thus,

$$
C^{-1}=\left(\begin{array}{rrr}
-1 & -1 & 1 \\
-2 & 0 & 1 \\
-2 & -1 & 1
\end{array}\right)
$$

Verify that $C^{-1} C=E$.

3. Using formula (1), find the column of coordinates of the vector $\mathbf{x}$ in the basis $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime}$:

$$
X_{\mathbf{e}^{\prime}}=C^{-1} X_{\mathbf{e}}=\left(\begin{array}{rrr}
-1 & -1 & 1 \\
-2 & 0 & 1 \\
-2 & -1 & 1
\end{array}\right) \cdot\left(\begin{array}{r}
1 \\
2 \\
3
\end{array}\right)=\left(\begin{array}{r}
0 \\
1 \\
-1
\end{array}\right)
$$

Answer. $\mathbf{x}_{\mathbf{e}^{\prime}}=\{0,1,-1\}$.

Conditions of the problem. Find the coordinates of the given vector $\mathbf{x}$ in the basis $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime}$.

1. $\mathbf{e}_{1}^{\prime}=\mathbf{e}_{1}+\mathbf{e}_{3}$,
2. $\mathbf{e}_{1}^{\prime}=\mathbf{e}_{1}$,
$\mathbf{e}_{2}^{\prime}=2 \mathbf{e}_{1}+\mathbf{e}_{2}+\mathbf{e}_{3}$
$\mathbf{e}_{2}^{\prime}=2 \mathbf{e}_{1}+\mathbf{e}_{2}$
$\mathbf{e}_{3}^{\prime}=\mathbf{e}_{2}$,
$\mathbf{e}_{3}^{\prime}=3 \mathbf{e}_{1}+2 \mathbf{e}_{2}+\mathbf{e}_{3}$
$\mathbf{x}_{\mathbf{e}}=\{3,-5,4\}$.
$\mathbf{x}_{\mathbf{e}}=\{1,2,7\}$.
3. $\mathbf{e}_{1}^{\prime}=\mathbf{e}_{1}$,
4. $\mathbf{e}_{1}^{\prime}=\mathbf{e}_{1}+\mathbf{e}_{2}+\mathbf{e}_{3}$,
$\mathbf{e}_{2}^{\prime}=\mathbf{e}_{1}+2 \mathbf{e}_{2}$
$\mathbf{e}_{2}^{\prime}=2 \mathbf{e}_{2}+2 \mathbf{e}_{3}$
$\mathbf{e}_{3}^{\prime}=\mathbf{e}_{1}+2 \mathbf{e}_{2}+3 \mathbf{e}_{3}$,
$\mathbf{e}_{3}^{\prime}=3 \mathbf{e}_{3}$,
$\mathbf{x}_{\mathbf{e}}=\{6,2,0\}$.
$\mathbf{x}_{\mathbf{e}}=\{2,6,6\}$.
5. $\mathbf{e}_{1}^{\prime}=3 \mathbf{e}_{1}+\mathbf{e}_{2}+5 \mathbf{e}_{3}$,
$\mathbf{e}_{2}^{\prime}=2 \mathbf{e}_{1}+3 \mathbf{e}_{2}+3 \mathbf{e}_{3}$,
$\mathbf{e}_{3}^{\prime}=2 \mathbf{e}_{1}+\mathbf{e}_{2}+4 \mathbf{e}_{3}$,
6. $\mathbf{e}_{1}^{\prime}=2 \mathbf{e}_{1}+\mathbf{e}_{2}+3 \mathbf{e}_{3}$,
$\mathbf{x}_{\mathbf{e}}=\{0,5,5\}$.
$\mathbf{e}_{2}^{\prime}=3 \mathbf{e}_{1}+2 \mathbf{e}_{2}+4 \mathbf{e}_{3}$,
$\mathbf{e}_{3}^{\prime}=2 \mathbf{e}_{1}-3 \mathbf{e}_{2}+\mathbf{e}_{3}$,
$\mathbf{x}_{\mathbf{e}}=\{9,14,16\}$.
7. $\mathbf{e}_{1}^{\prime}=2 \mathbf{e}_{1}+6 \mathbf{e}_{2}+5 \mathbf{e}_{3}$,
$\mathbf{e}_{2}^{\prime}=5 \mathbf{e}_{1}+3 \mathbf{e}_{2}-2 \mathbf{e}_{3}$,
$\mathbf{e}_{3}^{\prime}=7 \mathbf{e}_{1}+4 \mathbf{e}_{2}-3 \mathbf{e}_{3}$,
8. $\mathbf{e}_{1}^{\prime}=3 \mathbf{e}_{1}+2 \mathbf{e}_{2}+3 \mathbf{e}_{3}$,
$\mathbf{x}_{\mathbf{e}}=\{1,0,-1\}$.
$\mathbf{e}_{2}^{\prime}=-4 \mathbf{e}_{1}-3 \mathbf{e}_{2}-5 \mathbf{e}_{3}$,
$\mathbf{e}_{3}^{\prime}=5 \mathbf{e}_{1}+\mathbf{e}_{2}-\mathbf{e}_{3}$,
$\mathbf{x}_{\mathbf{e}}=\{-2,0,1\}$.
9. $\mathbf{e}_{1}^{\prime}=2 \mathbf{e}_{1}+3 \mathbf{e}_{2}+\mathbf{e}_{3}$,
$\mathbf{e}_{2}^{\prime}=7 \mathbf{e}_{1}+9 \mathbf{e}_{2}+5 \mathbf{e}_{3}$,
$\mathbf{e}_{3}^{\prime}=3 \mathbf{e}_{1}+4 \mathbf{e}_{2}+3 \mathbf{e}_{3}$,
10. $\mathbf{e}_{1}^{\prime}=\mathbf{e}_{1}+2 \mathbf{e}_{2}+2 \mathbf{e}_{3}$,
$\mathbf{x}_{\mathbf{e}}=\{0,3,3\}$.
$\mathbf{e}_{2}^{\prime}=2 \mathbf{e}_{1}+\mathbf{e}_{2}-2 \mathbf{e}_{3}$,
$\mathbf{e}_{3}^{\prime}=2 \mathbf{e}_{1}-2 \mathbf{e}_{2}+\mathbf{e}_{3}$,
$\mathbf{x}_{\mathbf{e}}=\{-9,0,9\}$.

Answers. 1. $\mathbf{x}_{\mathbf{e}^{\prime}}=\{5,-1,-4\}$. 2. $\mathbf{x}_{\mathbf{e}^{\prime}}=\{4,-12,7\}$. 3. $\mathbf{x}_{\mathbf{e}^{\prime}}=\{5,1,0\}$. 4. $\mathbf{x}_{\mathbf{e}^{\prime}}=\{2,0,0\}$. 5. $\mathbf{x}_{\math