Revision 54911be35993d9d31d1345a22ad09c656b773eb9 (click the page title to view the current version)
3D Mathematics
Reading Ma (2004) Chapter 2 + Appendix A
Briefing
- Recap Change of Basis
- New Representations of 3D Motion
- (More on 3D Motion)
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
- Exercise 2.1 a+d. (See Definition A.12 page 446.)
- Exercise 2.3. (See Definition A.13 page 447.)
- Exercise 2.6.
- Exercise 2.7.
- Exercise 2.10.
- 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
Continue on 3D Mathematics
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.