# Create data in terms of Y & X
    Y  <- c(4,7,8)
    X1 <- c(1,2,3)
    X0 <- rep(1, times=length(X1))
     
# Create the design matrix X
    X <- cbind(X0,X1)  # bind 2 column vectors together
    head(X)
 
#-1 X' X matrix
    t(X) %*% X

#-2 inverse(X' X) matrix
    inverse(t(X) %*% X)
    
#-3 beta_hat = (X'X)^-1 X' Y
    bhat <- inverse(t(X) %*% X) %*% t(X) %*% Y
    bhat      

#-4 Yhat = H Y
    Yhat <- X %*% inverse(t(X) %*% X) %*% t(X) %*% Y
    c(Yhat) 
    
# Compare to estimates from lm() 
    y.x1 <- lm(Y ~ X1)
    y.x1
    y.x1$fit