Linear combination of linear combinations?

[bonfire09]

When the book says "Members of [] are linear combinations of linear combinations of members of S". basically means the span of the members in subspace S. Since
= {c1s1 +...
+ cnsn|c1...cnεR and s1...snεS} what does [] mean? does it mean a linear combination of atleast one linear or more linear combinations from ?

 

[Fredrik]

Let V be a finite-dimensional vector space. For all subsets S⊂V, is the set of all v in V such that v is equal to a linear combination of members of S. Since that makes a subset of V, the definition applies to as well. So [] is the set of all v in V such that v is equal to a linear combination of members of .

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 []=, but if you look at the other two statements, it is.

Two alternative notations for : span S, $\bigvee S$.

 

[bonfire09]

Yeah I forgot to mention []=. But the book states that [] is a linear combination of linear combinations of the members of . 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 . And [] 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]

Yeah I forgot to mention []=.

I thought that was what you were asking about, not something you already knew.

But the book states that [] is a linear combination of linear combinations of the members of

You mean "of S", right? (Not "of ").

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 .

What does that mean? And did you again mean S when you wrote ?

And [] 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 [] is given by the definition of the [] notation.

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

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).

 

[blue_raver22]

In mathematics, a linear combination is a combination of two or more vectors, each multiplied by a scalar and then added together. In this context, a linear combination of linear combinations means that the vectors in can be expressed as a combination of vectors from , which are themselves linear combinations of vectors from . Essentially, this means that the vectors in can be written as a linear combination of other vectors in . [] refers to the set of all possible linear combinations of vectors from , which is known as the span of . Therefore, when the book states that "members of [] are linear combinations of linear combinations of members of S," it means that any vector in [] can be expressed as a linear combination of vectors from .