solve_linear_equation_system_20181116.R

# https://stackoverflow.com/questions/53337864/solve-a-linear-equation-system-b-0-rstudio
# Would like to solve a linear equation system like this:
# x_1*3+x_2*4+x_3*5+x_4*6+x_5,1*2=0
# x_1*21+x_2*23+x_3*45+x_4*37*+x_5,1*0=0
# x_1*340+x_2*24+x_3*25+x_4*31+x_5,1*0=0
# x_1*32+x_2*45+x_3*5+x_4*6+x_5,2*2=0
# x_1*9+x_2*11+x_3*13+x_4*49+x_5,2*0=0
# x_1*5+x_2*88+x_3*100+x_4*102+X_5,3*2=0

# [x_1] [x_2] [x_3] [x_4] [,5]
# [1,]   3    4     5     6     2
# [2,]  21   23    45    37     0
# [3,]  340  24    25    31     0
# [4,]  32   45     5     6     2
# [5,]    9  11    13    49     0
# [6,]   5   88   100   102     2

# Use the Singular Value Decomposition (SVD), if all elements of the diagonal part of the SVD are non zero
# only the trivial solutions exists
solution_space <- function(A){
  my_svd <- svd(A)
  if(all(my_svd$d != 0)){
    return(rep(0, ncol(A)))
  } else {
    return(my_svd$u[,my_svd$d == 0, drop=F])
  }
}

A %*% solution_space(A)

# You can try the code with these matrices:

A <- matrix(c(1,1,0,1,1,0,0,0,1), 3, 3)
A <- matrix(c(1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1), 4, 4)
A <- matrix(c(1,1,0,1,1,0,0,0,0), 3, 3)