Reading Ma (2004) Chapter 2 + Appendix A

Briefing

1. Change of Basis

Exercises

Note. Exercises in parentheses are optional. Please skip these unless you have a lot of time.

All exercises are from Ma 2004 page 38ff

1. Exercise 2.1 a+d. (See Definition A.12 page 446.)
2. Exercise 2.3. (See Definition A.13 page 447.)
3. Exercise 2.6.
   
4. Exercise 2.7.
   
5. Exercise 2.10.
   
6. Exercise 2.11. To calculate eigenvalues and -vectors in Python, you can use numpy.linalg.eig.

If this is too much for one session, we will continue with next week.

Debrief

Solution drafts

Exercise 2.1

Definition of a linear function

\[f(C\cdot X) = C\cdot f(X)\]

\[f(X+Y) = f(X) + f(Y)\]

If we take a function \(f(X) = X\cdot A\cdot X\) we can test if this is true, e.g.

\[f(X+Y) = (X+Y)A(X+Y) = (XA+YA)(X+Y) = XAX + XAY + YAX + YAY = f(X) + XAY + YAX + f(Y)\]

In this case we get two terms which we should not have had if the function were linear, and they are not zero in the general case. Hence this function is not linear. Similar calculations can be made for the function in the exercise.