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The problem asks to write the vector \(\begin{bmatrix} 3 \\ -5 \\ -3 \end{bmatrix}\) as a linear combination of the given vectors:

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

This means we need to find scalars \(c_1\), \(c_2\), and \(c_3\) such that:

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

This gives us the following system of linear equations by matching the components of the vectors:

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

Let me solve this system of equations to find \(c_1\), \(c_2\), and \(c_3\).The solution to the system of equations gives the following values for the scalars:

\[
c_1 = -\frac{37}{28}, \quad c_2 = \frac{17}{7}, \quad c_3 = \frac{33}{14}
\]

Thus, the vector \(\begin{bmatrix} 3 \\ -5 \\ -3 \end{bmatrix}\) can be written as a linear combination of the given vectors as:

\[
\begin{bmatrix} 3 \\ -5 \\ -3 \end{bmatrix} = -\frac{37}{28} \begin{bmatrix} -4 \\ 4 \\ 2 \end{bmatrix} + \frac{17}{7} \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix} + \frac{33}{14} \begin{bmatrix} -2 \\ -4 \\ 5 \end{bmatrix}
\]

Would you like further details on how this solution was derived, or have any other questions?

Here are 5 related questions:
1. How do you check if a vector can be written as a linear combination of other vectors?
2. What is the geometric interpretation of a linear combination of vectors?
3. How do you solve systems of linear equations using matrices?
4. What is the rank of a matrix, and how does it relate to linear combinations?
5. Can the process of finding linear combinations be automated for larger systems?

**Tip:** When solving systems of linear equations, using matrix methods like Gaussian elimination can be much faster for larger systems.

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

Learn how to write the vector [3, -5, -3] as a linear combination of the vectors [-4, 4, 2], [1, 4, -5], and [-2, -4, 5]. This problem involves solving a system of linear equations and understanding vector spaces, suitable for undergraduate students.

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