Number of Elements in Basis Sets for V: Prove It!

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Homework Help Overview

The problem involves proving that any two basis sets for a vector space V have the same number of elements. The context is within linear algebra, specifically focusing on the properties of basis sets and dimensions of vector spaces.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Some participants express uncertainty about the intricacy of the proof, with one suggesting that the statement may not hold true for finite dimensions. Others discuss the definitions of finite dimensionality and the properties of spanning sets and independent sets.

Discussion Status

The discussion is exploring different interpretations of the problem. Some participants are questioning the assumptions regarding the nature of basis sets and their dimensions, while others are attempting to outline necessary conditions for the proof.

Contextual Notes

There is a mention of confusion regarding the agreement on the proof's complexity, and the discussion includes considerations about the definitions of finite dimensional spaces and the characteristics of spanning sets.

FourierX
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Homework Statement



Prove:

Any two basis sets for V have the same number of elements.

Homework Equations





The Attempt at a Solution



Sounds obvious but is quite intricate to prove it.
 
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Is it? I don't agree, at least if the dimension V is finite.
 
You confused me even more :(
 
He said he did not agree that the proof is quite intricate.

I know, it always confuses me when people don't agree with me, too.

A space is said to be finite dimensional if and only if there exist a finite spanning set. In that case, since the number of vectors in a spanning set is an integer, there must exist a smallest spanning set. Since a basis is a set of vectors that is both a spanning set and independent you need to prove:

1) The smallest spanning set is independent. (Show that if it were not a independent, you could remove one of the vectors and still have a spanning set, contradicting the fact that it is smallest.)

2) No set of independent vectors can have more members than the smallest spanning set. (Take a supposedly independent set with more vectors and rewrite each in terms of the smallest spanning set.)
 

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