Interpret $Rank(T+U)\leq Rank(T)+Rank(U).$

Question

In linear algebra,I encountered a theorem which states that for two matrices $m\times n$ namely $A,B$ the following holds,
$Rank(A+B) \leq Rank(A)+Rank(B)$.Similarly if we say in terms of linear transformations $T,U:\mathbb F^n \to \mathbb F^m$ then $Rank(T+U)\leq Rank(T)+Rank(U)$.How to understand it geometrically,I have proved the result in terms of matrices formally,but I want to understand geometrically in terms of linear transforamtions why this should hold i.e. I want to get myself convinced geometrically.

Answer 1

Hint: Everything in the image of $T+U$ is a linear combination of elements from the image of $T$ and the image of $U$.