1. Aside: projections onto vectors
If X is an np data matrix and a a p1 vector then Y= Xa is the projection of X onto a
Values of Y give the coordinates of each observation along the vector a
Example: x1=(1, 2), x2=(2, 1), x3=(1,1) a=(3,  4)/5, So
Multivariate Data Analysis

2. Aside: projections onto vectorsx1 2  x3 x2  
(0,0)  1
Multivariate Data Analysis

3. Aside: projections onto vectorsx1 2  x3 x2  
(0,0)  1
a=(3,  4)/5
Multivariate Data Analysis

4. Aside: projections onto vectorsx1 2  x3 x2   y1
(0,0)  1
a=(3,  4)/5
Multivariate Data Analysis

5. Aside: projections onto vectors
x1=(1, 2) 2  x3 x2  
y1 = (3142)/5 = 1
(0,0)  1
a=(3,  4)/5
Multivariate Data Analysis

6. Aside: projections onto vectorsx1 2  x3 x2  
y1 = 1
(0,0)  1
y2 = 2/5
a=(3,  4)/5
Multivariate Data Analysis

7. Aside: projections onto vectorsx1 2 
y3 = 7/5 x3 x2  
y1 = 1
(0,0)  1
y2 = 2/5
a=(3,  4)/5
Multivariate Data Analysis

8. Aside: projections onto vectors2
y3 = 7/5  
y1 = 1
(0,0)   1
y2 = 2/5
a=(3,  4)/5
Multivariate Data Analysis

9. Aside: projections onto vectors2     
(0,0)   1
a=(3,  4)/5
Multivariate Data Analysis