Sum of Subspaces

Summation

$W_1+W_2=\{a+b:a\in W_1,b\in W_2\}$

Theorem

• Given $W_1,W_2\subset V$ are subspaces
• Then, their sum $W=W_1+W_2\subset V$ is also a subspace

Proof

1. Let $W_1,W_2$ be arbitrary vector spaces
2. To prove that $W_1+W_2$ is non-empty:
   1. Pick $\overrightarrow{a}\in W_1$
   2. Pick $\overrightarrow{b}\in W_2$
   3. Then, $a+b\in W_1+W_2$
   4. Thus, $W_1+W_2$ is non-empty
3. Applying Subspace Test
   1. Pick $\overrightarrow{x},\overrightarrow{y}\in W_1+W_2$ and pick $c\in F$
   2. We can write: $\overrightarrow{x}={\overrightarrow{x}}_1+{\overrightarrow{x}}_2$
   3. We can write: $\overrightarrow{y}={\overrightarrow{y}}_1+{\overrightarrow{y}}_2$
   4. Then. for $x_1,y_2\in W_1$ and $\overrightarrow{x_2},\overrightarrow{y_2}\in W_2$
   5. Consider $c\overrightarrow{x}+\overrightarrow{y}=c({\overrightarrow{x}}_1+{\overrightarrow{x}}_2)+({\overrightarrow{y}}_1+{\overrightarrow{y}}_2)$
   6. $=c({\overrightarrow{x}}_1+{\overrightarrow{y}}_1)+c({\overrightarrow{x}}_2+{\overrightarrow{y}}_2)$
   7. Note that $c({\overrightarrow{x}}_1+{\overrightarrow{y}}_1)\in W_1$
   8. Note that $c({\overrightarrow{x}}_2+{\overrightarrow{y}}_2)\in W_2$
   9. Then since $W_1$ and $W_2$ are subspaces, $c\overrightarrow{x}+\overrightarrow{y}\in W_1+W_2$
   10. Hence, $W_1+W_2$ is a subspace as they pass the Subspace Test