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The given task asks to express the vector \( \vec{x} = (-2, 4, 7)^T \) in terms of the basis vectors \( \vec{e}_1 = (0, 1, 2)^T \), \( \vec{e}_2 = (1, 0, 1)^T \), and \( \vec{e}_3 = (-1, 2, 4)^T \).

This means we need to find scalars \( a_1 \), \( a_2 \), and \( a_3 \) such that:
\[
\vec{x} = a_1 \vec{e}_1 + a_2 \vec{e}_2 + a_3 \vec{e}_3.
\]

The system of equations can be written as:
\[
\begin{pmatrix}
-2 \\
4 \\
7
\end{pmatrix}
=
a_1 \begin{pmatrix}
0 \\
1 \\
2
\end{pmatrix}
+ a_2 \begin{pmatrix}
1 \\
0 \\
1
\end{pmatrix}
+ a_3 \begin{pmatrix}
-1 \\
2 \\
4
\end{pmatrix}.
\]

This leads to the following system of equations for each component:

\[
\begin{aligned}
\text{1st component:} \quad & 0a_1 + 1a_2 - 1a_3 = -2, \\
\text{2nd component:} \quad & 1a_1 + 0a_2 + 2a_3 = 4, \\
\text{3rd component:} \quad & 2a_1 + 1a_2 + 4a_3 = 7.
\end{aligned}
\]

We can solve this system of linear equations to find the values of \( a_1 \), \( a_2 \), and \( a_3 \). I'll calculate these now.The solution to the system of equations is:

\[
a_1 = 2, \quad a_2 = -1, \quad a_3 = 1.
\]

Thus, the vector \( \vec{x} = (-2, 4, 7)^T \) can be written as:
\[
\vec{x} = 2 \vec{e}_1 - \vec{e}_2 + \vec{e}_3.
\]

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What are the general steps for expressing a vector in terms of a different basis?
2. How can we check if a set of vectors forms a basis for a vector space?
3. What is the geometric interpretation of expressing a vector in terms of a basis?
4. How does this method relate to solving systems of linear equations?
5. What are some practical applications of changing the basis of a vector?

**Tip:** When solving linear systems, it’s helpful to verify your results by substituting the values back into the original equations.

Написати розклад вектора x = (-2, 4, 7)^T за векторами e1 = (0, 1, 2)^T, e2 = (1, 0, 1)^T, e3 = (-1, 2, 4)^T.

Learn how to decompose the vector x = (-2, 4, 7)^T using the basis vectors e1, e2, and e3. This involves solving a system of linear equations to find the coefficients of the linear combination. This problem is suitable for students studying undergraduate linear algebra.

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