linear combination of linear combinations?


by bonfire09
Tags: combination, combinations, linear
bonfire09
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#1
Jan25-13, 08:44 PM
P: 219
When the book says "Members of [[S]] are linear combinations of linear combinations of members of S". [S] basically means the span of the members in subspace S. Since
[S] = {c1s1 +... + cnsn|c1...cnεR and s1...snεS} what does [[S]] mean? does it mean a linear combination of atleast one linear or more linear combinations from [S]?
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Fredrik
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Jan26-13, 01:52 AM
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Let V be a finite-dimensional vector space. For all subsets S⊂V, [S] is the set of all v in V such that v is equal to a linear combination of members of S. Since that makes [S] a subset of V, the definition applies to [S] as well. So [[S]] is the set of all v in V such that v is equal to a linear combination of members of [S].

It's possible to prove that if E,F⊂V, the following statements are equivalent (i.e. they're either all true or all false).

(a) E is the set of all v in V such that v is a linear combination of members of F.
(b) E is the intersection of all subspaces that have F as a subset.
(c) E is the smallest subspace that has F as a subset. (This means that if E' is a subspace that has F as a subset, E⊂E').

A set E for which these statements are true is, in your notation, denoted by [F]. If you only look at (a), it's not obvious that [[S]]=[S], but if you look at the other two statements, it is.

Two alternative notations for [S]: span S, ##\bigvee S##.
bonfire09
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#3
Jan26-13, 10:12 AM
P: 219
Yeah I forgot to mention [[S]]=[S]. But the book states that [[S]] is a linear combination of linear combinations of the members of [S]. Does this let's say set R={c1,1s1+...+cn,1sn,...,c1,m s1+...+cn,m sn} which is the set of all linear combinations of [S]. And [[S]] just means taking a linear combination of those members in R such as
r1(c1,1s1+...+cnsn)+...+rn(c1,m s1+...+c n,m sn) ? Oh s1,...sn. are elements of S and S is a subspace of V

Fredrik
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Jan26-13, 10:56 AM
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linear combination of linear combinations?


Quote Quote by bonfire09 View Post
Yeah I forgot to mention [[S]]=[S].
I thought that was what you were asking about, not something you already knew.

Quote Quote by bonfire09 View Post
But the book states that [[S]] is a linear combination of linear combinations of the members of [S]
You mean "of S", right? (Not "of [S]").

Quote Quote by bonfire09 View Post
Does this let's say set R={c1,1s1+...+cn,1sn,...,c1,m s1+...+cn,m sn} which is the set of all linear combinations of [S].
What does that mean? And did you again mean S when you wrote [S]?

Quote Quote by bonfire09 View Post
And [[S]] just means taking a linear combination of those members in R such as
r1(c1,1s1+...+cnsn)+...+rn(c1,m s1+...+c n,m sn) ?
I don't understand what you're asking, but the meaning of [[S]] is given by the definition of the [] notation.

Maybe you meant to ask this: If we define R=[S], does that make [[S]]=[R]? The answer is of course yes. The [[]] notation isn't something entirely different from the [] notation. It's just the [] operation done twice.

Quote Quote by bonfire09 View Post
S is a subspace of V
It doesn't have to be.

I recommend that you start using LaTeX, or at least vBulletin's sup and sub tags. (Like this: E=mc2).


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