Prove that the Gaussian filter:
is linear.
I'm not really sure how to approach this..
If I understand correctly, in order for it to be linear, we need that:
G(ax+z,y) = aG(x,y) + G(z,y)
However, I cant see how it actually stands for both additivity and homogeneity, as any of these (additivity/homogeneity) would change the exponent of the outcome only.
This question probably needs more contextual information to be answered properly. Since the function in the original post is a 2-dimensional Gaussian, and since it is referred to by the word "filter", I will assume the task originates from a context related to image processing, where the operation of "Gaussian filtering" essentially means "convolution with a 2D Gaussian kernel".
Under the above assumptions, it is sufficient to prove that, in general, the convolution operator is linear, i.e. $$(\alpha f+ \beta h)*g = \alpha (f*g) + \beta(h*g)$$ where $\alpha,\beta\in \mathbb{R}$ and $f,g,h:\mathbb{R}^2 \rightarrow \mathbb{R}$ satisfy the conditions for the existence of the convolution integral. Then simply set $g=G$.