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 $\begin{bmatrix}1&-2&3\\5&-9&h\\-3&6&-9\end{bmatrix}\sim\begin{bmatrix}1&-2&3\\0&1&h-15\\0&0&0\end{bmatrix}$.
Since the third column of the coefficient matrix contains no pivot for any value of $h$, we conclude that the homogeneous equation always contains a free variable, i.e., that all values of $h$ leave the columns linearly dependent.
