How are vector space dimension and basis related? (I am new to these concepts and know little to nothing about linear algebra/advanced calculus.) Thank you in advance.
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2The dimension of a vector space is the minimal number of elements needed to specify any vector in the space. So the number of elements in a basis = the dimension of the VS. – james h Aug 31 '15 at 00:03
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Every vector space has a basis. (Another proof). In particular, there exists a collection of vectors such that every vector can be written as a finite linear combination of vectors in the collection. We say that such a collection of vectors spans the vector space.
If there is a finite collection of vectors that spans a vector space, we say the vector space is finite-dimensional. It is perhaps not immediately obvious, but the minimum number of elements of a collection of vectors that span the vector space is well-defined. This is usually proved through the Steinitz Exchange Lemma (another).
We call the minimum number of vectors necessary to span a vector space the dimension of the vector space.
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