Assignment 8 (Nov 19 - Nov 26)

Reading: Sections 3.8, 3.9, 4.1, 4.2, 4.4
(we are skipping 4.3 until later)

Extra:  There may be some online questions for extra credit coming up from time to time.

Study Suggestions:

Section 3.8 and 3.9

Both sections are about Least Squares solutions using the Normal Equation ATAx = ATy.  You should understand how to solve for x* and for y* (to use the bookÕs notation) in general and how to apply this to fitting data to lines, parabolas, etc.  In addition, you are expected to understand that y* is the orthogonal projection of y to the Range of A, why AT(y-y*) = 0, and other key ideas.

DO some odd problems 3.9, 1-9.

Section 4.1 Eigenvalue Problem for 2x2

This section is really a preview of the whole chapter:  (1) what is an eigenvalue? (2) How can you find eigenvalues using determinants and the characteristic polynomial (3) once you have the eigenvalues, find the eigenvectors (4) what can happen if the characteristic polynomial has multiple or complex roots?  The more you can understand the flow of ideas in 4.1, the better you will understand the rest of the chapter.

DO: A sampling of the odd problems 1 – 11.

Section 4.2 Determinants and the Eigenvalue Problem

This section repeats the beginning of 4.1 but this time for nxn matrices.  The key is to become competent at finding determinants using the cofactor expansion.  Get the plus and minus signs straight. Pay special attention to shortcuts when there are loots of zeroes in the matrix or when the matrix is triangular.  It is true that in the general nxn case the determinant computations get too complicated for hand computation but you can do small sizes of n and some special cases.

DO: Problem 7.  Also 13, 15, 17

Section 4.4 Eigenvalues and the Characteristic Polynomial

This section uses the determinants from 4.2 study eigenvalues and to extend the definition of the characteristic polynomial to the nxn case.  There is no general root-finding formula for finding roots for degree n polynomials, but you can guess in some cases and use technology in others.  At this point it becomes important to think about the meaning of solution methods and not just grind them out.

DO: Problems 7, 9

Homework 8 (due Wednesday, 11/26 at the start of class – or earlier)

Technology note:  You can and should use a calculator or a computer for some of these problems.  But you should still show intermediate steps (matrix products and cofactor expansion) and not just write down the answer.  The idea is that one should be able to follow your thinking and the partial results but you can compute matrix products and solve linear equations using tech.

  1. Section 3.8, #12
  2. Section 3.9, #2
  3. Section 3.9, #8
  4. Section 4.1, #2, 4
  5. Section 4.1 #6
  6. Section 4.2 #14 do this by hand using cofactors; check with calculator
  7. Section 4.2 #16 do this by hand using cofactors; check with calculator
  8. Section 4.2 #18 do this by hand using cofactors; check with calculator
  9. Section 4.4 #8
  10. Section 4.4 #11