1.  Double descent.

2.  

0 correlation:  all
independent:  square, rectangle

3.  

simpolycorr <- function(x,deg)                                                  
{                                                                               
   d <- matrix(x,ncol=1)                                                        
   for (i in 2:deg) {                                                           
      d <- cbind(d,x^i)                                                         
      crr <- cor(d)                                                             
      crr <- crr[row(crr) != col(crr)]  # trick to get nondiag elements         
      print(max(crr))                                                           
      print(min(crr))                                                           
   }                                                                            
}                                                                               
                                                                                
simpolycorr(mlb$Age,6)                                                          
                                 

4.  

myPCA <- function(x)                                                            
{  cvx <- cov(x)                                                                
   eigout <- eigen(cvx)                                                         
   list(sdev=sqrt(eigout$values), rotation=eigout$vectors)                      
}                                                                               
                                                                                
library(regtools)
data(mlb)
mlbx <- mlb[,c('Height','Weight','Age')]                                        
myPCA(mlbx)                                                                     
pcout <- prcomp(mlbx)                                                           
pcout$sdev                                                                      
pcout$rotation