### Week 8 -- Sections 5.2-5.4

#### Short-term objectives

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

#### Long-term objectives

- 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)
- Understand the variety of existence and uniqueness phenomena that can
occur for
*n* linear equations in *p* unknowns.
(Sec. 5.4)