Week 6:  March 15-19

 

 Chapter 5.4   Least squares and data fitting

            Key points:       a)  Definition of least squares ‘solution’ to Ax = b.

                                    b)  Least squares x* is solution to ATAx = ATb.

                                    c)  It is crucial to explain least squares data fitting

            Homework:  10, 20, 22, 30, 36, 38*, 40*

 

 Chapters 6.1 and 6.2   Determinants

Key points:       a)  The reason why we are interested in determinants.  Thus verify

the 2-d assertion that a matrix is invertible if and only if its determinant is non-zero. 

b)  Give intuition as to why one should expect that there is a single

number that will decide invertibility in terms of the volume of a cube under the resulting linear transformation.  Assert, but don’t prove that this volume is the determinant.

                                    c)  The determinant is the product of the diagonals for upper-

      triangular matrices

d)  Row/Column operations don’t change the determinant (and why)

e)  Det(A) ≠ 0 if and only if A is invertible—proof uses row/column

f)  Det(A) = Det(AT).

g)  Det(AB) = Det(A)Det(B).

            Homework:  52 in Ch. 6.1, then 4, 14, 16, 26, 34*, 37* in Ch. 6.2

 

 Chapter 7.1  Introduction to eigenvectors and eigenvalues

            Key points:       a)  The notion of an eigenvector and eigenvalue are crucially

     important. 

b)  Use a dynamical system to motivate them

c)  A non-invertible matrix must have zero as an eigenvector

            Homework:  8, 20, 24, 34, 36, 7*, 38*