Subspace Addition: Understanding the Union of A and B

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

moonbeam said:
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.
 
Last edited:
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}.
 
Last edited:

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