To determine if the given vectors u, v, and w are linearly independent or dependent, we need to check if a linear combination of the vectors:

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

has only the trivial solution $c_1 = c_2 = c_3 = 0$.

The vectors are:

$\mathbf{u} = \begin{pmatrix} -2 \\ 0 \\ -3 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} -1 \\ -1 \\ 4 \end{pmatrix}, \quad \mathbf{w} = \begin{pmatrix} -4 \\ 2 \\ -17 \end{pmatrix}$

This can be tested by forming a matrix $A$ with u, v, and w as columns, and determining its determinant. If the determinant of the matrix is non-zero, the vectors are linearly independent. If the determinant is zero, they are linearly dependent.

The matrix $A$ is: