Write a function forwardSubstitution that takes three input arguments in this order: a lower triangular matrix $L,$ a permutation matrix $P,$ and a column vector $b.$ The function's output argument should be a column vector $c$ comprising the solution to the equation $Lc=Pb.$ Check that the input matrices and vector have compatible sizes and return a value of $[]$ if they are incompatible.

Write a function backSubstitution that takes an upper triangular matrix $U$ and a column vector $c,$ in that order, as input. The output should be a column vector $x$ that is the solution to the equation $Ux=c.$ Check that the input matrix and vector have compatible sizes and return a value of $[]$ if they are incompatible.

DO NOT USE COLON NOTATION

USE LOOPS