Proving W=V When W is a Subspace of an n-Dimensional Vector Space

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Homework Help Overview

The discussion revolves around proving that if W is a subspace of an n-dimensional vector space V and dim(W) = n, then W = V. The subject area is linear algebra, specifically focusing on concepts related to vector spaces and subspaces.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definitions of 'dim' and 'subspace', with some questioning how to illustrate the relationship between W and V. There are discussions about linear combinations, bases, and the implications of dimension in vector spaces.

Discussion Status

Some participants have provided insights into the relationship between linear independence, spanning sets, and dimension, suggesting that understanding these concepts is crucial for addressing the problem. Multiple interpretations and approaches are being explored without a clear consensus.

Contextual Notes

There is a mention of needing to reference prior material on linear independence and dimension, indicating that foundational knowledge may be necessary for the discussion.

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Homework Statement



Prove that if W is a subspace of an n-dimensional vector space V and dim(W) = n, then W=V


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The Attempt at a Solution



I don't know where to start.
 
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What does 'dim' mean, and what does subspace mean? Can you look it up? That should get you started.
 
Last edited:
I understand why W would equal V. Every linear combination of vectors in W is in W, and since W and V have the same number of elements in a basis, and W is in V, then W=V. I just don't know how to illustrate that.
 
Last edited:
Probably the text, just before this, has some results about "linear independence", "spanning sets", and "dimension". Probably there is a proof that shows any two bases for a given vector space are the same size. That material is what you need to understand in order to do this problem.
 
As g edgar suggested, and I will make even more specific:

If any two of these are true, so is the third
1) There exist a set of n independent vectors
2) There exist a set of n vectors that span the space
3) The dimension is n
 

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