If $A=B+C$, here $A,B,C$ are semi-positive definite matrices, prove that $\text{im}A=\text{im}B+\text{im}C$.
It is trivial to prove $\text{im}A\subset\text{im}B+\text{im}C$. For the other side, a vector in $\text{im}B+\text{im}C$ can be written as $Bx+Cy$. And we derive the following: \begin{align} \\xAx=xBx+xCx \\yAy=yBy+yCy \end{align}
Our goal is to find $z$ such that $Az=Bx+Cy$, but I have no ideas how to use the positive condition.
We will use the property that if $T\ge 0$ then $$\|Tx\|\le \|T\|\langle Tx,x\rangle \quad (*)$$ Since $A$ is semi-positive definite then $Ax=0$ implies $Bx=Cx=0.$ Indeed if $A=0$ then $$0=\langle Ax,x\rangle = \langle Bx,x\rangle + \langle Cx,x\rangle $$ Hence $$\langle Bx,x\rangle= \langle Cx,x\rangle=0$$ By $(*)$ we get $Bx=Cx=0.$ The implication $$Bx=Cx=0\implies Ax=0$$ is obvious. Therefore $$\ker A=\ker B\cap \ker C$$ Taking the orthogonal complement gives $${\rm Im}\, A={\rm Im}\, B+{\rm Im} \,C$$