1.

A <- rbind(c(3,4),c(1,-1),c(4,3))
y <- c(18,-1,17)
MASS::ginv(A) %*% y


2.

library(rectools)
hv <- getHouseVoting()
hv <- hv[,-17]
library(softImpute)
hv <- as.matrix(hv)
hvic <- as(hv,'Incomplete')
aEst <- softImpute(hvic,5)
si <- softImpute(hvic,5)
# could use complete() instead:
aEst <- si$u %*% diag(si$d) %*% t(si$v)
aEst[3,1]  # 0.35

3.

library(MASS)

als <- function(m,r,nmf=FALSE,
   w0=matrix(runif(nrow(m)*r),nrow=nrow(m)),
   eps=0.01*max(abs(m)),maxIters=100) 
{
   w <- w0
   h <- matrix(rep(0,r*(ncol(m))),nrow=r)

   for (iter in 1:maxIters) {
      oldmest <- w %*% h
      gw <- ginv(w)
      for (j in 1:ncol(m)) {
         h[,j] <- gw %*% m[,j]
         if (nmf) h[,j] <- pmin(h[,j],0)  # not min*(
      }
      gh <- ginv(t(h))
      for (i in 1:nrow(m)) {
         ## w[i,] <- t(ginv(t(h)) %*% m[i,])
         w[i,] <- t(gh %*% m[i,])
         if (nmf) w[i,] <- pmin(w[i,],0)  # not min*(
      }
      if (frob(w %*% h - oldmest) < eps) break
      if (iter == maxIters) warning('did not converge')
   }
   list(w=w,h=h,prd=w %*% h)

}

frob <- function(mat) 
{
   sqrt(sum(mat^2))
}