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I've just started working with abstract algebra, and while the theory makes some sense, I have a bit of trouble figuring out the actual methods to complement the theory.

For example, a base for a space is a set which spans the set, and is linearly independent. I know how to show lineare independence (solve equation and confirm that the solution set is $\{0, 0, \ldots, 0\}$)$\ldots$ but how does one show that a set "spans" a space?

There are a few examples in the book where it just says "these two vectors span the space because I can write any vector in the space as a linear combination with these two; obviously", but how would one show this when working with a base that doesn't "obviously" span the space?

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There are two main approaches. One is to take an arbitrary nonzero vector, say $(a,b)$ in $\Bbb R^2$ and show you can express it in terms of your basis. The second is to know the theorem that if you have as many vectors as the dimension of the space and they are linearly independent, they span the space. This only works for finite dimensional spaces.

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This method usually works for me:

Figure out how many of the vectors in the set are linearly independent (using the method that you have stated in the question)

Suppose there are m n dimensional independent vectors in the set (m <= n, as the number of independent vectors can't exceed the dimension of the vectors).

The set will span an m dimensional subspace of R$^n$, but it doesn't have to be R$^m$

Example:

You have two 3 dimensional independent vectors. The set will span a plane in R$^3$, but it doesn't have to be R$^2$

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