Linear algebra, Projection

Question

Let $Proj_\vec{u}(\vec{v})$ be the projection of $\vec{u}$ where $\vec{u},\vec{v}\in R^2$. If we fixed $\vec{u}$ and let $\vec{v}$ be a variable, then how can we see $Proj_\vec{u}(\vec{v})$ as a function in, $$Proj_\vec{u}(\vec{v}):R^n\to R^n$$ Do you think $Proj_\vec{u}(\vec{v})$ is a real linear function with respect to $\vec{v}$?