Answer
$h=6$
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 $\begin{bmatrix}1&3&-1\\-1&-5&5\\4&7&h\end{bmatrix}\sim\begin{bmatrix}1&3&-1\\0&1&-2\\0&0&h-6\end{bmatrix}$.
If there is a pivot in every column of the coefficient matrix, then the homogeneous system has only the trivial solution. To avoid this, we can set $h=6$ so that the final row is all zeros.
