Quick Solution

Given:

\( \vec{a} = \hat{i} - \hat{k}, \quad \vec{b} = x\hat{i} + \hat{j} + (1 - x)\hat{k}, \quad \vec{c} = y\hat{i} + x\hat{j} + (1 + x - y)\hat{k} \)

Form the matrix:

\( M = \begin{bmatrix} 1 & x & y \\ 0 & 1 & x \\ -1 & 1 - x & 1 + x - y \end{bmatrix} \)

Find the determinant:

\( \det(M) = \begin{vmatrix} 1 & x & y \\ 0 & 1 & x \\ -1 & 1 - x & 1 + x - y \end{vmatrix} = 1 \)

Since the determinant is constant and non-zero, the vectors are linearly independent.

\( \boxed{\text{The matrix does not depend on } x \text{ or } y} \)