9/23/2008

Finding the image

Properties of the image

• The zero vector in R^n is in the image of T.
• The image of T is closed under addition: If v1 and v2 and in the image of T, then so is v1 + v2.
• The image of T is close under scalar multiplication: If v is in the image of T and k is an arbitrary scalar, then kv is in the image of T as well.

Definition of kernel

Properties of the kernel

• The zero vector in R^m is in the kernel of T.
• The kernel is closed under addition
• The kernel is closed under scalar multiplication.

Various characterizations of invertible matrices

For an n x n matrix A, the following statements are equivalent; that is, for a given A they are either all true or all false.

• A is invertible.
• The linear system Ax = b has a unique solution x, for all b in R^n.
• rref(A) = I_n.
• rank(A) = n.
• im(A) = R^n.
• ker(A) = {0}.

Subspaces of R^n

Linear Independence

Redundant vector

Basis of a subspace

Group/Class Exercises

• Section 3.1 - 1, 5, 11, 17, 19
• Section 3.2 - 15, 26

Please read Section 3.3 for class on Thursday