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.