Finding kernel of linear transformation

Suppose I have a vector space V generated by {x1x2x3,x32,x21x4} over Q and another vector space W generated by {x21x3,x1x22} over Q and L is linear transformation which maps x1x2x3↦x21x3, x32↦3x1x22 and x21x4↦x21x3.

Then using Mathematica how can I find the kernel space of this linear transformation.