1. Matlab tutorial and Linear Algebra Review Today’s goals: Learn enough matlab to get started. Review some basics of Linear Algebra Essential for geometry of points and lines. But also, all math is linear algebra. (ok slight exaggeration). Many slides today adapted from Octavia Camps, Penn State.

2. Vectors Ordered set of numbers: (1,2,3,4) Example: (x,y,z) coordinates of pt in space.

3. Vector Addition v w V+w

4. Scalar Product v av

5. Operations on vectors sum max, min, mean, sort, … Pointwise:.^

6. Inner (dot) Product v w  The inner product is a SCALAR!

7. Matrices Sum: A and B must have the same dimensions

8. Matrices Product: A and B must have compatible dimensions Identity Matrix:

9. Matrices Transpose: If A is symmetric

10. Matrices Determinant: A must be square

11. Matrices Inverse: A must be square

12. Indexing into matrices

13. Euclidean transformations

14. 2D Translation t P P’

15. 2D Translation Equation P x y txtxtxtx tytytyty P’ t

16. 2D Translation using Matrices P x y txtxtxtx tytytyty P’ t tP

17. Scaling P P’

18. Scaling Equation P x y s.x P’ s.y

19. Rotation P P’

20. Rotation Equations Counter-clockwise rotation by an angle  P x Y’ P’  X’ y

21. Degrees of Freedom R is 2x2 4 elements BUT! There is only 1 degree of freedom:  The 4 elements must satisfy the following constraints:

22. Stretching Equation P x y S x.x P’ S y.y

23. Stretching = tilting and projecting (with weak perspective)

24. Linear Transformation SVD

25. Affine Transformation

26. Files

27. Functions Format: function o = test(x,y) Name function and file the same. Only first function in file is visible outside the file.

28. Conclusions Quick tour of matlab, you should teach yourself the rest. We’ll give hints in problem sets. Linear algebra allows geometric manipulation of points. Learn to love SVD.