- 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.

- 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.

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}.

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