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?

Question

amistre64 · Answer

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.

amistre64 · Answer

$$span\begin{pmatrix}1&0&3\0&1&2\end{pmatrix}$$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.

amistre64 · Answer

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.