Quiz 3

Problem 1: The Rank and Nullity Theorem.

1) Find a basis for the rowspace of A.

2) Find a basis for the nullspace of A.

3) What is the rank of A?

4) What is the nullity of A?

5) The rank + the nullity of A is...?

6) Find a basis for the rowspace of A^T.

7) Find a basis for the nullspace of A^T.

8) The Rank + the nullity of A^T is ...?

Problem 2: The Column Space of A and A^T

1) Find a basis for the rowspace of A. Find a basis for the column space of A^T. Are they the same?Explian any differences.

2) Find a basis for the rowspace of A^T. Find a basis for the column space of A. Are they the same? Explain any differences.

Problem 3: The Change of Basis Matrix

Let S = { v₁, v₂, v₃} and T = { w₁, w₂, w₃} be bases for R³, where

and

1) Find the transition matrix from T to S.

2) If

what are a, b, c for

T

S