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bonfire09

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- Thread starter bonfire09
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In summary, the book says that members of S are linear combinations of linear combinations of members of S.

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bonfire09

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Fredrik

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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 [

Two alternative notations for

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bonfire09

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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

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Fredrik

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

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

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 [

Maybe you meant to ask this: If we define R=

It doesn't have to be.bonfire09 said:S is a subspace of V

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

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blue_raver22

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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

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.

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.

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.

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.

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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