Week 8 -- Sections 5.2-5.4

Short-term objectives

1. Find the kernels and ranges of simple linear functions. (Sec. 5.2)
2. Determine whether a linear function is injective, or surjective, or an isomorphism. (Sec. 5.2)
3. Prove the elementary theorems about kernels and ranges and the superposition principles. (Secs. 5.2, 5,3)
4. Distinguish between homogeneous and inhomogeneous linear equations. (Sec. 5.3)
5. Use superposition principles in solving linear equations. (Sec. 5.3)
6. Determine the rank of a matrix (or of a linear function representable by a matrix). (Sec. 5.4)
7. Use the theorem relating rank and nullity. (Sec. 5.4)
8. For a given matrix shape, give an example matrix of each possible rank. (Sec. 5.4)

Long-term objectives

1. Understand how the concepts of kernel, range, and affine subspace are useful in solving linear equations (especially ordinary and partial differential equations). (Secs. 5.2 ("Important Remarks") and 5.3)
2. Understand the variety of existence and uniqueness phenomena that can occur for n linear equations in p unknowns. (Sec. 5.4)