Math 423 - Final Exam Review
General Information
The final exam (Wednesday, May 8, 1-3 pm) will have 6 to 8 questions,
some with multiple parts, and will explicitly cover sections 3.7,
5.4-5.9, 6.1-6.4, 7.1-7.4, 7.5 (unitary matrices), and my class notes. Please bring two
8½×11 bluebooks. Problems will be similar to ones
done for homework. You may use calculators to do arithmetic, although
you will not need them. You may not use any calculator that
has the capability of doing either calculus or linear algebra.
Topics Covered
- For sections 3.7, 5.4-5.8, see the Test I
review; and for sections 5.9, 6.1-6.4, 7.1-7.4, 7.5, see the Test II review. There will be no proofs from
this material.
- Selfadjoint matrices
- Know and be able to prove
Theorems i and ii in the Notes.
- Be able to able to find the principal axes of a quadratic form , and for
two-variable forms be able to identify the type of conic given by the
level curves of the form.
- Singular value decomposition
- Know what the SVD is and be
able to find it for a given matrix. Also, be able to sketch a proof
for existence of the SVD.
- We mentioned several applications of the SVD. Be able to use it
to find the 2-norm condition number of an invertible matrix and also
to use it in solving least
squares problems and in obtaining the pseudoinverse of a matrix.
- Finite element
method
- Understand the finite element example.
- Know what linear splines are.
- Non-orthogonal least
squares come up in finite element problems, and in many other
applications as well. Be able to prove the lemma about the connection
between the invertibitiy of a Gram matrix and the linear indepence of
the vectors that generate it.
- Triangular forms of matrices
- Be able to get the Schur form of a matrix. Know what the block
triangular and Jordan canonical forms are.