(From Axler)Why does these two proof of these questions not contradict to each other?

Question

Here are the solutions to these 2 questions:

http://linearalgebras.com/3A.html

(Axler 3A 10) Suppose $U$ is a subspace of $V$ with $U≠V$ Suppose $S∈L(U,W)$ and $S≠0$ (which means that $Su≠0$ for some $u∈U$). Define $T:V→W$ by

$Tv$=$Sv$ if $v∈U$ and $Tv$=$0$ ,if $v∉U$

Prove that $T$ is not a linear map on $V$.

11. Suppose $V$ is finite-dimensional. Prove that every linear map on a subspace of $V$ can be extended to a linear map on $V$. In other words, show that if $U$ is a subspace of $V$ and $S∈L(U,W)$, then there exists $T∈L(V,W)$ such that $Tu=Su$ for all $u∈U$.

I know the theorem in Axler 3.5 and I can understand and agree with the proof. But the question 10 contradicts question 11 since the map in 11 is proved to be not linear in question 10.

I wonder where is wrong or I misunderstood something. Could someone explain the subtle difference?

Answer 1

The map in question 11 is not the same as the one in question 10. In question 10, the map has to be zero for all $v$ outside of $U$, whereas this is not required of the map in question $11$.