Addition of Subspaces

  • Thread starter moonbeam
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  • #1
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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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  • #2
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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.
 
  • #3
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Ok, subspaces of [tex]\mathbb{R}^3[/tex] have the following properties: contain the zero vector, are closed under addition, and are closed under multiplication. Am I right?
So, say [tex]A[/tex], [tex]B[/tex], and [tex]C[/tex] are subspaces of [tex]\mathbb{R}^3[/tex]. Then, what would [tex](A+B) \cap C[/tex] mean?
 
  • #4
Hurkyl
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Ok, subspaces of [tex]\mathbb{R}^3[/tex] have the following properties: contain the zero vector, are closed under addition, and are closed under multiplication. Am I right?
So, say [tex]A[/tex], [tex]B[/tex], and [tex]C[/tex] are subspaces of [tex]\mathbb{R}^3[/tex]. Then, what would [tex](A+B) \cap C[/tex] mean?

As per the definition of intersection, [itex](A+B) \cap C[/itex] is the set of all vectors that are both in [itex]A+B[/itex] and in [itex]C[/itex].
 
  • #5
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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 [tex]A[/tex], [tex]B[/tex], and [tex]C[/tex] are subspaces of [tex]\mathbb{R}^3[/tex]. Then, what would [tex](A+B) \cap C[/tex] mean?

It would mean that you have in your hands a subspace of R^3.
 
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  • #6
radou
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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 [itex]\in[/itex] A, b [itex]\in[/itex] B}.
 
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