Math 308 Review 3
The final will cover all the topics of this quarter from
Chapters 1, 2, 3, 4, 6 and 8. Below is a review of the topics after the second
midterm, so Sections 8.1, 8.2 and 8.5.
Terminology
Dot product, orthogonal vectors, orthogonal sets, a vector
orthogonal to a subspace, orthogonal complement of a subspace, orthogonal subspaces,
orthogonal basis, orthonormal basis, norm, distance between vectors, projection
of a vector onto a vector, projection of a vector onto a subspace, Gram-Schmidt, least squares
regression, normal equations for Ax=y.
Operations
1.
Taking the dot
product of two vectors.
2.
Finding the norm
(length) of a vector.
3.
Projecting one vector
onto another.
Applications
1.
Checking if two
vectors are orthogonal and checking if a vector is orthogonal to a subspace
using its spanning set or basis.
2.
Given a basis for a
subspace, finding an orthogonal basis for the same subspace: Gram-Schmidt
process.
3.
Projecting a vector
onto a subspace (requires an orthogonal basis for the subspace)
4.
Finding the least
squares solution to Ax=y using Ax=z where z is the projection of y onto the column space of A.
5.
Finding the least
squares solution to Ax=y using the normal
equations.
Theory Review
When you answer the true/false questions in
the book, make sure you can justify your answer. If you answer true, explain by
finding the relative theorem in the book. If you answer false, come up with a counter
example. With projection questions, it may help to sketch pictures. No matter
which space they live in, two independent vectors span a plane (the paper),
three independent vectors span a space (use some perspective on the paper to
give it a 3D look).
More questions
1. The textbook has a lot of questions with
the answers to the odd numbered ones at the back.
2. Kristen DeVleming’s exam archive
http://www.math.washington.edu/~kdev/teaching/308exams/
3. You can also look at some of the
questions from Linear
Algebra Done Wrong. Specifically,
p. 128, Exercises 2.1 and 2.2. (The range
is the column space)
p.134, Exercises 3.1, 3.5, 3.7
p. 141, Exercise 4.4.