1. Some Important Subspaces (10/29/04) If A is an m by n matrix, then the null space of A is the set of all vectors x in R n which satisfy the homogeneous matrix equation A x = 0. Denoted Nul(A). The null space of A is a subspace of R n (why??). Example: If A is n by n and invertible, what is Nul(A)?

2. Column Spaces If A is an m by n matrix, then the column space of A is the set of all linear combinations of the columns of A. Denoted Col(A) The column space of A is a subspace of R m (why?). Example: If A is n by n and invertible, what is Col(A)?

3. Linear Transformations If V and W are any vector spaces, then a function T with domain V and codomain W is a linear transformation provided that it preserves (or “respects”) the operations in V and W, i.e., If u and v are vectors in V, then T (u + v) = T (u) + T (v) If u is in V and c is a scalar, then T (c u) = c T (u) This is “old hat” for us, just more general.

4. Range Old term: The range of a linear transformation T (from V to W ) is the set of all vectors in W of the form T (x) for some x in V. The range of T is a subspace of W (why?). Note that the range of T is simply the column space of T ‛s standard matrix if T goes from R n to R m.

5. Kernel New term: The kernel of a linear transformation T (from V to W ) is the set of all vectors x in V for which T (x) = 0 The kernel of T is a subspace of V (why?). Note that the kernel of T is simply the null space of T ‛s standard matrix if T goes from R n to R m.

6. Assignment for Monday Read Section 4.2. Do the Practice and Exercises 1, 3, 5, 7, 9, 17, 21, 25, 29, 31. On Monday, you will be given Hand-In Homework #2, which will be due next Friday afternoon (not at class time). There will be no class on Wed (11/3).