# Linear combination of linear combinations?

• bonfire09
In summary, the book says that members of S are linear combinations of linear combinations of members of S.
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 ?

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##.

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

bonfire09 said:
Yeah I forgot to mention []=.

bonfire09 said:
But the book states that [] is a linear combination of linear combinations of the members of

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

bonfire09 said:
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 ?

bonfire09 said:
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.

bonfire09 said:
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).

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 .

## 1. What is a linear combination of linear combinations?

A linear combination of linear combinations is a mathematical operation that involves combining multiple linear combinations to form a new linear combination. It is commonly used in linear algebra to simplify equations and solve systems of linear equations.

## 2. How is a linear combination of linear combinations calculated?

To calculate a linear combination of linear combinations, you must first determine the coefficients of each linear combination. Then, you multiply each coefficient by its respective linear combination and add all of the results together.

## 3. What is the purpose of using a linear combination of linear combinations?

The main purpose of using a linear combination of linear combinations is to simplify and manipulate complex equations. It allows for easier computation and can help in solving systems of equations. It is also used in applications such as data analysis and machine learning.

## 4. Can a linear combination of linear combinations be applied to vectors?

Yes, a linear combination of linear combinations can be applied to both scalars and vectors. In fact, it is commonly used in linear algebra for vector operations such as matrix multiplication and solving systems of linear equations.

## 5. Are there any limitations to using a linear combination of linear combinations?

While a linear combination of linear combinations is a powerful tool in linear algebra, it is not always applicable in all situations. For example, it may not be useful in nonlinear systems or when dealing with non-numeric data. Additionally, the resulting equations may become too complex to solve by hand, requiring the use of computational tools.

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