#LINEAR COMBINATIONS OF VARIABLES
#USE BUILT-IN iris DATA:
iris
#LINEAR COMBINATIONS OF VARIABLES
#USE BUILT-IN iris DATA:
iris
M=subset(iris,select=c("Sepal.Length","Sepal.Width",
   "Petal.Length","Petal.Width"))
M

#LINEAR COEFFICIENTS:
a=c(2,3,-1,5)
a
b=c(0,2,1,4)
b

#LINEAR COMBINATIONS:
lca=t(a)%*%t(M)
lca
lcb=t(b)%*%t(M)
lcb

#STATISTICS OF ORIGINAL VARIABLES:
Xbar=mean(M)
Xbar
S=cov(M)
S

#STATISTICS OF LINEAR COMBINATIONS:
#MEAN
lcabar=mean(lca)
lcbbar=mean(lcb)
#VARIANCE
lcavar=var(t(lca))
lcbvar=var(t(lcb))
#COVARIANCE
lccov=cov(t(lca),t(lcb))
Result=c(lcabar,lcbbar,lcavar,lcbvar,lccov)
Result

#IDENTITIES DEMONSTRATED:
t(a)%*%Xbar
t(a)%*%S%*%a
t(a)%*%S%*%b

#MULTIPLE LINEAR COMBINATIONS:
A=read.table("c:/DATA/Multivariate/LC Matrix A.txt",header=F)
A=as.matrix(A)
LC=as.matrix(A)%*%t(as.matrix(M))

#IDENTITIES DEMONSTRATED FOR MULTIPLE LINEAR COMBINATIONS:
#MEAN:
LC=data.frame(t(LC))
LCbar=mean(LC)
LCbar
A%*%Xbar
#COVARIANCE MATRIX:
cov(LC)
A%*%S%*%t(A)