TITLE: The definition of direct sums and subspaces?
QUESTION [15 upvotes]: From what I have read the definition of the direct sum of the vector spaces $V_1$, $V_2$ is the set $V_1\times V_2$ with the operations of addition and scalar multiplication defined as follows (Knapp, 2006):
\[(u_1,u_2)+(v_1,v_2)=(u_1+v_1,u_2+v_2)\]
\[c(v_1,v_2)=(cv_1,cv_2)\]
But then today I came across the fact that if $V$ has linear subspaces $V_1$, $V_2$ and every $v\in V$ can be written uniquly as:
$$v=v_1+v_2$$
for $v_i \in V_i$ then $V=V_1\oplus V_2$. I cannot see how this makes sense given the definition above. Is the above definition wrong? If not how can $V$ have the form $(v_1,v_2)$ when $v_1,v_2 \in V$?

REPLY [6 votes]: I have just found this pdf: http://www.math.ncku.edu.tw/~fjmliou/advcal/sumvspace.pdf which summarizes direct sums, and explains what is going on here. I will here outline what it says. Thanks also to the other answers which helped clarify things.
This is to do with the difference between an internal and external direct product:
The External Direct Sum
Let $V$ and $W$ be two vector spaces over $F$ then we the external direct product $V\oplus_e W$ is defined as I have given it in the question. i.e. the vector space with underlying set $V\times W$ and operators defined by:
\[(u_1,u_2)+(v_1,v_2)=(u_1+v_1,u_2+v_2)\]
\[c(v_1,v_2)=(cv_1,cv_2)\]
The Internal Direct Sum
For $V_1$ and $V_2$ subspaces of $V$ which satisfy:

for each $v\in V$ there exists $v_1 \in V_1$ and $v_2 \in V_2$ such that $v=v_1+v_2$.
$V_1 \cap V_2\{0\}$

in this case we write $V=V_1 \oplus_i V_2$ and $V$ is the inner direct product of $V_1$ and $V_2$.
Link between them
There is a linear isomorphism such that $V_1\oplus_e V_2$ is isomorphic to $V_1 \oplus_i V_2$