The R package pracma provides the functions null and nullspace to compute to vectors spanning the kernel of a matrix mapping. Its name comes from the alternative name "null space" for "kernel". For the computation of the image of a linear mapping you can use the function orth.

> library(pracma)
> A <- matrix(c(1, 0, 1, 2, 1, 1, -1, 1, -2), nrow=3, byrow=TRUE);  A
     [,1] [,2] [,3]
[1,]    1    0    1
[2,]    2    1    1
[3,]   -1    1   -2
> null(A)        # kernel of A
           [,1]
[1,] -0.5773503
[2,]  0.5773503
[3,]  0.5773503
> Rank(A)        # rank of A
[1] 2
> orth(A)        # spanning vectors of the image of A
           [,1]          [,2]
[1,] -0.4264014  3.330669e-16
[2,] -0.6396021 -7.071068e-01
[3,]  0.6396021 -7.071068e-01
> round(.Last.value, 4)
        [,1]    [,2]
[1,] -0.4264  0.0000
[2,] -0.6396 -0.7071
[3,]  0.6396 -0.7071