## Spanning Sets

Example: Take the vector space as the set of linear polynomials of degree 1.

A possible spanning set is

\[\{1,x \}\]

.With respect to this spanning set

\[2-x=2(1)+(-1)(x)\]

which we may write \[\begin{pmatrix}2\\-1\end{pmatrix}\]

.Another possible spanning set is

\[ \{1-x,1+x \}\]

.In terms of this spanning set

\[2-x=(1-x)+(1+x)-\frac{3}{2}(1-x) + \frac{1}{2}(1+x)\]

which we may write as the vector \[\begin{pmatrix}3/2\\1/2\end{pmatrix}\]

Every vector space has an associated dimension

\[n\]

and every set of elements has an order \[m\]

. There are four possible cases.If

\[m the set of elements does not span the vector space. If

There are four possible cases.

If

\[m=nn\]

the set of elements forms a basis for the vector space if and only if the set of elements of \[S\]

are linearly independent. If the set of elemtns of \[S\]

are linearly dependent then we can express at least one of the elements of \[S\]

in terms of other elemts, so throw out this element to get a small set with \[n>m\]

so \[S\]

is not a spanning set.There are four possible cases.

If

\[m>n\]

the set of elements is not linearly independent and may or may not span the vector space. We can reduce the siz of \[S\]

by expressing vectors in terms of other vectors, throwing out the vectors so expressed until we reduce \[S\]

to a linearly independent set with order \[s\]

. If