TITLE: Difference between sum and direct sum
QUESTION [42 upvotes]: What is the difference between sum of two vectors and direct sum of two vector subspaces? 
My textbook is confusing about it. Any help would be appreciated.

REPLY [18 votes]: You need to keep in mind these two definitions: 

Definition 1: sum of subsets
Suppose $U_1,...,U_m$ are subsets of $V$. The sum of $U_1,...,U_m$
  denoted by $U_1+...+U_m$ is the set of all possible sums of elements
  $U_1,...,U_m$. More precisely, 
$$U_1 + ... + U_m = \{u_1+...+u_m: u_1\in U_1,...,u_m\in U_m\}$$
Definition 2: direct sum
Suppose $U_1,...,U_m$ are subspaces of $V$. 
The sum $U_1+...+U_m$ is called a direct sum if and only if each
  element of $U_1+...+U_m$ can be written in one and only one way as
  a sum of $u_1+...+u_m$ where each $u_j$ is in $U_j$. The notation for
  such a sum of subsets is $U_1\oplus ... \oplus U_m$.

There's a shortcut to identify direct sums:
Suppose $U_1,...,U_m$ are subspaces of $V$. Then $U_1+...+U_m$ is a direct sum if and only if the only way to write $0$ as a sum $u_1+...+u_m$ is by taking each $u_j$ equal to $0$.
Also, 
Suppose $U$ and $W$ are subspaces of $V$. Then $U+W$ is a direct sum if and only if $U\cap W =\{0\}$.