To determine if the vectors $\mathbf{u} = \begin{pmatrix} -2 \\ 0 \\ -3 \end{pmatrix}$, $\mathbf{v} = \begin{pmatrix} -1 \\ -1 \\ 4 \end{pmatrix}$, and $\mathbf{w} = \begin{pmatrix} -4 \\ 2 \\ -17 \end{pmatrix}$ are linearly independent or dependent, we need to check whether there exists a nontrivial linear combination of these vectors that equals the zero vector.

This can be expressed as:

$c_1 \mathbf{u} + c_2 \mathbf{v} + c_3 \mathbf{w} = \mathbf{0}$

In matrix form, this equation becomes:

$\begin{pmatrix} -2 & -1 & -4 \\ 0 & -1 & 2 \\ -3 & 4 & -17 \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$

To determine whether the vectors are linearly independent, we will check the determinant of the matrix formed by the vectors $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$. If the determinant is non-zero, the vectors are linearly independent; if the determinant is zero, the vectors are linearly dependent.

Let me compute the determinant.

$\text{det} \left( \begin{pmatrix} -2 & -1 & -4 \\ 0 & -1 & 2 \\ -3 & 4 & -17 \end{pmatrix} \right)$

I will now calculate the determinant.The determinant of the matrix is $0$, which means the vectors $\mathbf{u} = \begin{pmatrix} -2 \\ 0 \\ -3 \end{pmatrix}$, $\mathbf{v} = \begin{pmatrix} -1 \\ -1 \\ 4 \end{pmatrix}$, and $\mathbf{w} = \begin{pmatrix} -4 \\ 2 \\ -17 \end{pmatrix}$ are linearly dependent.

Would you like further details or have any questions?

Here are some follow-up questions for further exploration:

1. How is the determinant related to the concept of linear independence?
2. Can you explain the geometric interpretation of linearly dependent vectors?
3. How would the solution change if we added a fourth vector?
4. What are some practical applications of checking linear independence in real-world problems?
5. How does row reduction help in determining linear independence?

Tip: When vectors are linearly dependent, at least one vector can be expressed as a linear combination of the others.