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## Homework Statement

V is a vector space

## The Attempt at a Solution

If S is smaller than V then there exists a T such that S + T = V. OTHERWISE S = V. I'm not sure what assumptions am I making which I could break down to prove...

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- Thread starter ashina14
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- #1

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V is a vector space

If S is smaller than V then there exists a T such that S + T = V. OTHERWISE S = V. I'm not sure what assumptions am I making which I could break down to prove...

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How do I show that the span of B is appropriate for T?

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if S is the empty space, the solution is obvious

if dim S >= 1 there is a base of vectors (ui) of S.

And there is a theorem who says that

the family of vectors (ui) can be completed

with a family of vectors (vj) so that

the union (ui) with (vj) is a basis of V

finally the subspace T generated by (vj) = (v1, v2, ...)

in the complementary space of S so that

V=S⊕T

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i'm not sure if i need to quote the theorem

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How do I prove that? I don't see an obvious connection here

- #8

Mark44

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It is against Physics Forums rules to post complete solutions. You have received numerous warnings about this, and each comes with a private message to you. Apparently you aren't reading your PMs so I am posting something to you in this thread.

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sorry'

Hedi

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Thanks for the help anyway :)

- #12

Mark44

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It is against Physics Forums rules to post complete solutions. You have received numerous warnings about this, and each comes with a private message to you. Apparently you aren't reading your PMs so I am posting something to you in this thread.

My post was addressed to hedipaldi, not you, ashina14. See above.

The rules are here: https://www.physicsforums.com/showthread.php?t=414380.

- #13

Mark44

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I'm glad to hear that!

sorry'

Hedi

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