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In my attempts to learn linear algebra from Khan Academy I've come across several concepts that I can't completely connect.

Every vector space has a base. This base consists of the minimal elements that can span the entire vector space.

The number of elements in this vector space is defined to be the vector space's dimension.

When trying to conclude the power of a vector space (as in how many vectors there are in the space), I know that $|V| = |F| ^ n$

I also know that this small n = dimension of V. What I don't understand is why does the dimension equals that $n$?

Thanks!

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    I feel this question is illuminated by the intuitive interpretation of $(1,0,\dots,0),(0,1,0,\dots,0),\dots ,(0,\dots,0,1)$ as a series of vectors (if you prefer, arrows) of length $1$ specifying direction in $n$ different directions. This makes the idea of dimension quite geometrically obvious.2012-12-04

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A basis for a vector space is a linearly independent set that spans the space. For the Euclidean space $\mathbb R^n$, a basis is $(1,0,...,0),(0,1,0,...,0),...,(0,...,0,1)$. Since there are $n$ elements in this set, the dimension of $\mathbb R^n$ is $n$.

Bases for a vector space are not unique. It can be shown that any two bases have the same number of elements, so that this dimension is well-defined.

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    Is there a particular name for this basis in your answer? Something like 'canonical vector space basis'?2012-12-04