9/25/2008

Problem Session

General Discussion

• What is the null space of A?
• What is the rank of a matrix?
• What is a linear relation?
• Do dependent vectors have a nontrivial relation?

The relationship between kernel and relations

Various characterizations of linear independence

For a list of vectors v1, v2, ... vm in R^n, the following statements are equivalent:

• The vectors are linearly independent.
• None of the vectors is redundant.
• None of the vectors is a linear combination of the other vectors in the list.
• There is only the trivial relation among the vectors
• ker [ v1 v2 ... vm ] = 0.
• rank [ v1 v2 ... vm ] = m.

Basis and Unique Representation

Group/Class Exercises

• Section 3.2 - 15, 26, 31, 33, 39, 52
• Section 3.3 - 5, 7

Please read Section 3.4 for class on Tuesday