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The problem is asking to express the vector \(\begin{bmatrix} 4 \\ -3 \\ -2 \end{bmatrix}\) as a linear combination of three given vectors:

\[
\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} -3 \\ 2 \\ 5 \end{bmatrix}, \quad \begin{bmatrix} 3 \\ 4 \\ -1 \end{bmatrix}
\]

We need to find scalars \(c_1\), \(c_2\), and \(c_3\) such that:

\[
\begin{bmatrix} 4 \\ -3 \\ -2 \end{bmatrix} = c_1 \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} + c_2 \begin{bmatrix} -3 \\ 2 \\ 5 \end{bmatrix} + c_3 \begin{bmatrix} 3 \\ 4 \\ -1 \end{bmatrix}
\]

This leads to the following system of equations by equating the components of the vectors:

1. \( 1c_1 - 3c_2 + 3c_3 = 4 \)
2. \( -1c_1 + 2c_2 + 4c_3 = -3 \)
3. \( 0c_1 + 5c_2 - 1c_3 = -2 \)

I will solve this system to find \(c_1\), \(c_2\), and \(c_3\).The solution to the system is:

\[
c_1 = \frac{44}{17}, \quad c_2 = -\frac{13}{34}, \quad c_3 = \frac{3}{34}
\]

Thus, the vector \(\begin{bmatrix} 4 \\ -3 \\ -2 \end{bmatrix}\) can be written as:

\[
\frac{44}{17} \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} - \frac{13}{34} \begin{bmatrix} -3 \\ 2 \\ 5 \end{bmatrix} + \frac{3}{34} \begin{bmatrix} 3 \\ 4 \\ -1 \end{bmatrix}
\]

Let me know if you'd like more details on the solution process!

### Related questions:
1. How do you check if a set of vectors is linearly independent?
2. What does it mean for vectors to form a basis?
3. Can any vector be written as a linear combination of other vectors?
4. How is the rank of a matrix related to linear combinations?
5. How does the concept of linear combinations relate to vector spaces?

#### Tip: When solving systems of equations, arranging them in matrix form can make the process clearer and help visualize the solution!

Write [4 -3 -2] as a linear combination of the vectors [1 -1 0], [-3 2 5], [3 4 -1].

Learn how to express the vector [4 -3 -2] as a linear combination of the vectors [1 -1 0], [-3 2 5], and [3 4 -1]. This problem involves solving a system of linear equations using the method of linear combinations. Suitable for advanced high school or college students.

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