- #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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- 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".

- #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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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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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

Homework Helper

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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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