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A set of vectors form a basis for vector space V if the set of vectors are lin independent AND span V. So after finding the span of V you can simply find the basis by taking away lin dependent vectors right?

Mathematics
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a basis is the most efficient span. a span can contain useless vectors in it; for example, take a plane. a plane only need to be defined by 2 independant vectors. If you have a span that contains more than 2 vectors that are coplanar, then the extra vectors are useless in defining a basis.
\[span\begin{pmatrix}1&0&3\\0&1&2\end{pmatrix}\]is not a basis even tho it spans R^2. The column vector [3,2] can be formed from the first 2 and therefore provides no extra benefit in determing any other vector in the vectorspace.
To find the basis of a given matrix A; row reduce it to B and remove all the columns in A that relate to "free variables". The rest of the column vectors of A will form a basis.

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