**Linear Independence:**-
are linearly
**independent**if has only the trivial solution. **Characterization:**-
are linearly

**dependent**if some is a linear combination of the other vectors. **Basis:**-
is a
**basis**for a subspace*H*of*V*if*B*is linearly independent and*Span*(*B*) =*H*.*R*^{n}:**standard basis**is .*P*_{n}:**standard basis**is .- Pivot columns for
*A*form a basis for*Col*(*A*). - Spanning set for solutions of
is a basis for
*Nul*(*A*).

**Spanning Set Theorem:**- Suppose
and
*H*=*Span*(*S*).- 1.
- Removal of some dependent
from
*S*gives a smaller spanning set for*H*. - 2.
- If
,
some subset of
*S*is a basis for*H*.

Assume a basis
for vector space *V*.

**Uniqness:**- for any , uniquely.
**Coordinates:**- if
,
then
the
*c*_{i}'s are the*B***coordinates**of ; notation: . **Change-of-Coordinates Matrix:**- For vectors in
*R*^{n}, the**change-of-coordinates matrix***P*_{B}is defined by ; .

2000-07-06