# Addition of Subspaces

I just wanted to know if subspace A + subspace B is the same as the "union of A and B".

## Answers and Replies

I just wanted to know if subspace A + subspace B is the same as the "union of A and B".

I never seen this notation. It does not really make sense because the union of two suspaces is never a subspace unless one is contained in the other. Perhaps, it means the set of all sums, each one from each subspace.

Ok, subspaces of $$\mathbb{R}^3$$ have the following properties: contain the zero vector, are closed under addition, and are closed under multiplication. Am I right?
So, say $$A$$, $$B$$, and $$C$$ are subspaces of $$\mathbb{R}^3$$. Then, what would $$(A+B) \cap C$$ mean?

Hurkyl
Staff Emeritus
Gold Member
Ok, subspaces of $$\mathbb{R}^3$$ have the following properties: contain the zero vector, are closed under addition, and are closed under multiplication. Am I right?
So, say $$A$$, $$B$$, and $$C$$ are subspaces of $$\mathbb{R}^3$$. Then, what would $$(A+B) \cap C$$ mean?

As per the definition of intersection, $(A+B) \cap C$ is the set of all vectors that are both in $A+B$ and in $C$.

I just wanted to know if subspace A + subspace B is the same as the "union of A and B".

Not in general.
But A+B always includes AUB.
In fact, span(AUB) = A+B.

So, say $$A$$, $$B$$, and $$C$$ are subspaces of $$\mathbb{R}^3$$. Then, what would $$(A+B) \cap C$$ mean?

It would mean that you have in your hands a subspace of R^3.

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As pointed out in the posts above, one only has to go through definitions: for two subspaces A, B of V, you have A + B = [A U B] = {a + b : a $\in$ A, b $\in$ B}.