# Sum of two subspaces is also a subspace

There is this Corollary that says:

Let $U_1$ and $U_2$ be subspaces of vector space $V$, then $U_1+U_2$ is also a subspace of $V$.

I looked for proof and i found this:

$$U_1=(x_1, y_1)$$

$$U_2=(x_2, y_2)$$

$$x=(x_1+x_2) x_1\in U_1, x_2\in U_2.$$

$$y=(y_1+y_2) y_1\in U_1, y_2\in U_2$$

$$x+y=x_1+x_2+y_1+y_2=(x_1+y_1)+(x_2+y_2) \in U_1+U_2$$

I know that subspaces are closed under addition but I can't relate it to what i found. Any explanation?

• Another approach would be to show that $U_1 + U_2$ is the image of the linear map $A : V \to V$ defined by $A(v) = P_{U_1}(v) + P_{U_2}(v)$, where $P_{U_1}$ and $P_{U_2}$ are projections onto $U_1$ and $U_2$ respectively. The image of a linear map is always a subspace. – Bungo Sep 26 '17 at 19:26

## 1 Answer

The proof you found is not complete. You have to check all axioms for a linear space.

The lines just show that $U_1 + U_2$ is closed under addition. Set $U=U_1 + U_2$ and let $a,b\in U$. We have to show $a+b\in U$. Since $a\in U$, there exists $a_1\in U_1$ and $a_2\in U_2$ such that $a=a_1 + a_2$. Similar for $b$. Hence we can write $$a+ b = a_1 + a_2 + b_1 + b_2.$$ Since $a_i, b_i \in U_i$ and the operation $+$ is commutative we further have $$a_1 + a_2 + b_1 + b_2 = a_1 + b_1 + a_2 + b_2$$ and since the $U_i$ are linear spaces, $a_1 + b_1 \in U_1$ and $a_2 + b_2 \in U_2$. But this shows $a+b \in U$.

All you have to do now is to carefully check the remaining axioms. You will see that the properties of $U_1$ and $U_2$ transfer to $U$.

Hint: Instead of checking all axioms you should use a so called subspace criterium/definition. See here.

• In the courses I have seen, it would have been considered inappropriate to check, e.g. associativity, since this shows that the standard subspace inheritance properties are not understood. Closure under addition and scaling (or something similar) is definitely the best route. – Zach Boyd Sep 26 '17 at 17:12