# [R] Finding solution set of system of linear equations.

Date: Sat, 21 May 2011 18:34:20 -0700 (PDT)

I have a simple system of linear equations to solve for X, aX=b: > a

[,1] [,2] [,3] [,4]
[1,] 1 2 1 1
[2,] 3 0 0 4
[3,] 1 -4 -2 -2
[4,] 0 0 0 0
> b

[,1]
[1,] 0
[2,] 2
[3,] 2
[4,] 0

(This is ex Ch1, 2.2 of Artin, Algebra). So, 3 eqs in 4 unknowns. One can easily use row-reductions to find a homogeneous solution(b=0) of:
X_1 = 0, X_2 = -c/2, X_3 = c, X_4 = 0

and solutions of the above system are:
X_1 = 2/3, X_2 = -1/3-c/2, X_3 = c, X_4 = 0.

So the Kernel is 1-D spanned by X_2 = -X_3 /2, (nulliity=1), rank is 3.

In R I use solve():
> solve(a,b)
Error in solve.default(a, b) :
Lapack routine dgesv: system is exactly singular

and it gives the error that the system is exactly singular, since it seems to be trying to invert a.
So my question is:
Can R only solve non-singular linear systems? If not, what routine should I be using? If so, why? It seems that it would be simple and useful enough to have a routine which, given a system as above, returns the null-space (kernel) and the particular solution.

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