Answer
all values of $h$
Work Step by Step
The vectors are linearly dependent if and only if the matrix equation $\mathbf{A}\vec{x}=\vec{0}$ has a nontrivial solution, where $\mathbf{A}$ is the matrix whose columns are the given vectors. By row reduction, we conclude that $\begin{bmatrix}2&-6&8\\-4&7&h\\1&-3&4\end{bmatrix}\sim\begin{bmatrix}1&-3&4\\0&0&h+16\\0&0&0\end{bmatrix}$.
Since the second column of the coefficient matrix contains no pivot for any value of $h$, we conclude that the solution to the homogeneous equation will always contain a free variable, i.e., that all values of $h$ leave the columns linearly dependent.
