Direct sums

Also read 1C. Subspaces

Definition:

Let Assume that $V_1,V_2,....,V_n$ are subspaces of $V$. Then we can define the elements in the Sum of Subspaces as $v_1+v_2+....+v_n$ for each $v_i\in V_i$

A direct sum are sums where every element can only be represented by $v_1+v_2+....+v_n$ in one way.

If $V_1+V_2+....+V_n$ is a direct sum it is denoted $V_1\oplus V_2\oplus ....\oplus V_n$.

#todo Prove the equality

Powerful condition to determine direct sum

A sum is a direct sum $⟺$ the only way to write the $0$ vector is by setting each summad to $0$

$\Rightarrow$ This is easy to prove because it is clear $0$ exists in every vector space, and the sum of all these 0 vectors is 0. By definition of direct sum this can be the only way to write 0 as it can only be represented by one such sum.

$\Leftarrow$ Assume the only way to write 0 is $0+0+.....+0$. Now we have to prove this the sum is a direct sum

1. For the sake of contradiction, assume the sum is not a direct sum. Thus there exists an element $v$ in the sum that can be written as $v_1+v_2+...+v_n$ and $w_1+w_2+...+w_n$ where each $v_i,w_i\in V_i$
2. Then notice $(v_1+v_2+...+v_n)-(w_1+w_2+...+w_n)=v-v=0$
3. This means $(v_1-w_1)+(v_2-w_2)+(v_3-w_3)+....+(v_n-w_n)=0$
4. As there is only one way to write 0, by setting each summand equal to 0, thus every $v_i+w_i=0⟹v_i=w_i$

Condition for direct sum of two subspaces

Suppose $V_1+V_2$ is a sum of two subspaces $V_1,V_2\in V$: $V_1+V_2\text{is a direct sum}⟺V_1\cap V_2=\{0\}$ Proof:

1. $V_1+V_2\text{is a direct sum}⟹V_1\cap V_2=\{0\}$
   1. Assume $V_1+V_2$ is a direct sum. For the sake of contradiction, assume there exists some $v\in V_1\cap V_2\ne 0$.
   2. Then, $v\in V_1,v\in V_2$ meaning that its inverse is also in both,$-v\in V_1\cap V_2$
   3. Then lets take $v\in V_1$ and $-v\in V_2$ and we have an element in the direct sum: $v+(-v)=0$
   4. However the only way to write zero is by setting all the summands to zero as $V_1+V_2$ is a direct sum. Thus $v,-v=0$ and the only element in the intersection is $0$ as any other element would have to be $0$
2. $V_1\cap V_2=\{0\}⟹V_1+V_2\text{is a direct sum}$
   1. Assume $V_1\cap V_2=\{0\}$