Section 1

week of August 31

Brief review of the linear algebra quiz
1. Follow the hint to actually determine all the entries in the matrix and check that it is symetric.
2. Writing out the formulas and matching them up is enough... Note that we could also have had a,b,c in the rows instead of the columns of A.
```
      | a_1  b_1  c_1 |                                       
      | a_2  b_2  c_2 |  =  a_1 b_2 c_3 - a_1 b_3 c_2         
      | a_3  b_3  c_3 |      + a_3 b_1 c_2 - a_2 b_1 c_3      
                                + a_2 b_3 c_1 - a_3 b_2 c_1   
                         =  a_1 (b_2 c_3 - b_3 c_2)           
                           +a_2 (b_3 c_1 - b_1 c_3)        
                           +a_3 (b_2 c_1 - b_1 c_2)        
                         = a [dot] (b [cross] c)           
```
3. For part (a) you actually need to verify the properties. The matrix for (b) looks like :
```
   0   -u_3   u_2 
```
```
  u_3    0   -u_1 
```
```
 -u_2   u_1    0 
```
4. Noticing that for any vector w, w'A'Aw = (Aw)' Aw which is just the dot product of Aw with itself and therefore positive, we then consider an eigenvector v. Since A'A is symmetric we know that v is real and we have v'A'Av = v' \lambda v = \lambda v'v. We know the expression is positive from above, and that v'v is positive, so the eigenvalue \lambda must be positive. Let ' mean the conjugate transpose.
Some examples of 2D transformations
- Write the (3x3) matrix (using homogeneous coords) to transform points A,B,C to points A', B', C'. Then find the matrix which takes A', B', C' to A", B", C". Then compose this matrix with the first matrix to get one which takes A,B,C to A",B",C". What is the structure of this matrix? The answers are below the figure.
  To transofrm A,B,C to A',B',C' :
```
       0  -1  15
       1   0   0  
       0   0   1 
```
  which is rotation followed by translation, but notice that when we compose the matricies we put them in the opposite order. This is because the matrix on the right acts on the points first, then the matrix on the left:
```
       0  -1  15       1   0  15       0  -1   0
       1   0   0   =   0   1   0   *   1   0   0
       0   0   1       0   0   1       0   0   1
```
  The matrix to take A',B',C' to A",B",C" is just the same rotation matrix from above:
```
        0  -1   0
        1   0   0
        0   0   1
```
  Composing this (on the LEFT) with the previous tranformation matrix gives:
```
     0  -1   0      0  -1  15       0  -1   0       
     1   0   0  *   1   0   0   =  -1   0  15   
     0   0   1      0   0   1       0   0   1       
```
  Notice that in this matrix the rotation part (the upper left 2x2 part of the matrix) is simply the rotation matrix for 180 degrees counter clockwise. This is the sum of the rotations in each transformation. The translation part though has been rotated by the second 90 degree rotation so that now the 15 is a translation in the y direction.
Classifying 2D transformations
- Rigid: Translation, Rotation, Reflection
- Translation and Rotation can be represented as a rotation possibly around a point at inifinity. Reflection cannot.
- Non-Rigid: Scale, Shear
- What commutes?
  - Two translations commute
```
 T1 * T2 = T2 * T1 
```
  - Two scales commute
```
 S1 * S2 = S2 * S1 
```
  - Two rotations sometimes commute. In 2D rotations do commute, while in 3d most pairs of rotations do not commute. This will make rotations in 3D somewhat more complex.
  - Rotations and translations do not commute.
```
 R * T != T * R 
```
  - Translations and scales do not commute.
```
 S * T != T * S 
```
  - Scales and rotations commute only in the special case when scaling by the same amount in all directions. In general the two operations do not commute.
  - For each of these it is probably worthwhile to go through a quick example in your head for each pair of operations, including the special cases.
Quiz: Write the 3x3 matrix to transform A=(0,0) B=(1,0) C=(0,1) to A'=(2,3) B'=(3,2) C'=(2,2).

Basically the idea was to find some series of transformations (eg tranlsate(0,-1) rotate(3pi/4 cw) scale(1/sqrt(2),1) shear(0.5,0) and translate(2,3) ) which could be combined to produce the final transformation matrix. The idea was to make you think about the various kinds of transformations and get your hands dirty.
```
  0   1   2 
```
```
 -1  -1   3 
```
```
  0   0   1 
```
Always remember to check your answers... For problems like this it is easy, just see if it transforms all the points to the correct locations.

Some of you may have noticed that three points define an affine transform (in 2D) and it is especially easy to find the transformation if the starting points are (0,0) (0,1) and (1,0). To do this consider the point A=(0,0) which must be transformed to (2,3). Since only the entries in the first and second row of the last column of the matrix will control the result of transforming (0,0) we know they must be (2,3). Then we use that (0,1) goes to (3,2) to solve for the entries in the first and second row of the second column. And similarly for the first column using that (1,0) goes to (2,2).

aberg@cs.berkeley.edu