In "Sheldon Axler's Linear Algebra Done Right, 3rd edition", on page 21 "internal direct sum", or direct sum as the author uses, is defined as such:
Let $U_1, U_1,...,U_m$ be subspaces of $V$ vector space over field $F$, if any $v\in V$ can be written uniquely as a sum of $u_1+u_2+...+u_m$ where each $u_i\in U_i$, then the sum ($U_1+U_2+...+U_m$) is internal direct sum. 
On page 23 following this definition, a condition to check whether or not a sum of subspaces is internal direct is stated as follows:
A direct sum is internal direct sum if and only if the only way to write $0$ vector as the sum of $u_1+u_2+...+u_m$ where each $u_i\in U_i$ is equal to $0$. 
Question (1): How do we infer  from ?
Question (2): Why does checking whether or not $0$ vector can be uniquely written as the sum of $u_1+u_2+...+u_m$ where each $u_i=0$ suffice to say that the sum is internal direct sum? In other words, why is it not possible to be able to write $0$ vector uniquely while other vectors $v\neq 0$ are not writable uniquely, or vice versa?
I checked out many books about Linear Algebra as to a rigorous explanation for that, but I could not find anything satisfactory. Being these said, what I look for is some eloquent explanation to all these questions I stated above.