Let $V$, $W$ be $F$-vector spaces such that $V$ has a direct sum decomposition $V= U_1+U_2$. Let $F_1: U_1 → W$ and $F_2: U_2 → W$ be two linear maps. We say $F : V → W$ is a common extension of $F_1$ and $F_2$ if $F$ agrees with $F_1$ on $U_1$ and $F$ agrees with $F_2$ on $U_2$. Show that there is a common extension $F : V → W$ of $F_1$ and $F_2$.
I am confused as to the idea of the direct sum decomposition. Does it just mean that $U_1$ and $U_2$ are linearly independent? And then if $F_1(U_1) = W$ and $F_2(U_2) = W$ then wouldn't $F(V)$ be $2W$? I am confused. Thank you for the help.
Direct sum means that we can write $V = U_1 \bigoplus U_2 $ where the only vector in common to $U_1, U_2$ is $0$.