1. Basis of a Vector Space (11/2/05)
We generalize the concept of linear independence to vector spaces:
Definition. A set of vectors {v1, v2, …, vn} in a vector space V is linearly independent if the vector equation c1 v1 + c2 v2 + … + cn vn = 0 has only the trivial solution c1=0, …, cn=0.
Otherwise, the set is called linearly dependent.

2. Examples in P3
Classify each set of vectors in P3 (all polynomials of degree 3 or less in the variable t ) as either linearly independent or linearly dependent:
{1, t , 2t + 3}
{t + 1, t – 1}
{1, t , t 2 , t 3}

3. Definition of Basis
If H is a subspace of a vector space V and B = {v1, v2, …, vn} is a set of vectors, then B is a basis for H provided:
B is linearly independent, and
B spans H .
Since V is a subspace of itself, the definition applies to V as well.
The idea is that a basis is a minimal spanning set .

4. Examples of bases
The vectors (1 ,1, 0), (0, 2, -1) and (1, 0, 3) are a basis for R3 (check).
Guess what the “standard basis” for R3 is.
The polynomials 3, 2t – 1, and t are a basis for P2 (check).
Guess what the “standard basis” for P2 is.
Hence, note, bases are not unique.

5. The Spanning Set Theorem
If the set of vectors {v1, v2, …, vn} spans a subspace H of a vector space and if one vector vk from the set is a linear combination of the others, then it can be removed from the set and the new smaller set will still span H .
If H  {0}, then some subset of our original set is a basis for H .

6. Moving Up or Down, and Assignment
So you can move down from a spanning set, removing dependent vectors, until you arrive at a basis, or you can move up from a linearly independent set, adding more independent vectors, until you arrive at a basis.
For Friday, please Read Section 4.3 and do Exercises 1, 3, 5, 7, 9, 13, 15, 19, 21, 23, 33.