How do I determine linearity for the operator $L = x\partial_x + y\partial_y$?
I know the the definition of a linear operator, but I'm not sure what to do with $\partial_x$ and $\partial_y$ and there doesn't seem to be any mapping?
Asked
Active
Viewed 50 times
0
-
There is a mapping from the function $f$ to another function $L(f)$ which the latter function is defined by $L(f)(x,y)=x\partial_{x}f(x,y)+y\partial_{y}f(x,y)$. – user284331 Jan 13 '18 at 00:35
1 Answers
1
\begin{align*} (x\partial_{x}+y\partial_{y})(f+g)&=x\partial_{x}(f+g)+y\partial_{y}(f+g)\\ &=x\partial_{x}(f)+x\partial_{x}(g)+y\partial_{y}(f)+y\partial_{y}(g)\\ &=(x\partial_{x}+y\partial_{y})(f)+(x\partial_{x}+y\partial_{y})(g), \end{align*} can you do the scalar multiple case?
user284331
- 55,591