1. What is Linear Algebra? Notation:

2. Linear Transformation Linear Operator Matrix Multiplication n-Dimensional Linear Mapping Linear Coordinate Transformation

3. Matrix Multiplication

4. Just a common kind of function Mappings from Input to Outputs dose response contrast firing rate column space row space

5. Linear Transforms: x1x1 x2x2 a

6. a Identity b

7. Linear Transforms: Stretch and Squash a b

8. Linear Transforms: Flips b a

9. Linear Transforms: Rotation b a

10. Linear Transforms: Skew a b

11. Linear Transforms: Any Combination of the Above b a

12. Linear Transforms: Mix and Match, Collect the Whole Set

13. Singular Value Decomposition: Any Linear Transform can be written as: a rotation, a stretch and flip, and another rotation

14. EigenVectors: Vat gözinta, cümzouta. ‘eigen’ is German for ‘self’

15. a Linear Transforms: Identity b

16. Linear Transforms: Flips b a

17. Linear Transforms: Rotation b a

18. But What’s this good for?

19. fMRI

20. Time series realignment

21. 3D Rigid-body Transformations A 3D rigid body transform is defined by: –3 translations - in X, Y & Z directions –3 rotations - about X, Y & Z axes The order of the operations matters TranslationsPitch about x axis Roll about y axis Yaw about z axis

22. Linear Regression

29. fMRI

30. Intensity Time Regression model = 11 22 + + error x1x1 x2x2   Ν   Ι  (error is normal and independently and identically distributed) Question: Is there a change in the BOLD response between listening and rest? Hypothesis test:  1 = 0? (using t-statistic) General case

31. Y=    X     X   (1,1,1) (x1, x2, x3) XX XX O Y1x111Y2=x21+2Y3x313Y1x111Y2=x21+2Y3x313 DATA (Y 1, Y 2, Y 3 ) Y design space Geometrical perspective

32. Linear Regression

34. What is Differential Equation? Let’s take this a bit further:

35. What is Differential Equation? Let’s take this a bit further: And you all remember linear systems:

36. Linear Dynamical Systems! Can you read this?

37. Phase Space: f1f1 f2f2 What happens if A is a stretcher?

38. Differential Linear Systems It is the nature of A which determines the behavior of the system

39. Differential Linear Systems The return of Eigenvectors...

40. Differential Linear Systems

41. Nonlinear Dynamical Systems