# Let `A` be an `n xx n` matrix such that `A^2=0` . Prove that the column space of `A` is a subspace of the null space of `A.`

degeneratecircle | Certified Educator

If we label the columns of `A` as `vec c_1, vec c_2,...,vec c_n,` the column space is just the span of these vectors (which is the same as the range of the linear transformation described by `A,` as we'll prove below.) In other words, the column space is span{`vecc_1,...,vecc_n`}.

Let `a_1vecc_1+a_2vecc_2+...+a_nvecc_n` be an arbitrary element in the column space. Note that

`a_1vecc_1+...+a_nvecc_n=[vecc_1,...,vecc_n][[a_1],[.],[.],[.],[a_n]]=Avecx,`

where `vecx` is the vector consisting of the ```a` 's in the linear combination on the left hand side. We wish to show that this vector `Avecx` is in the null space of `A,` but that now follows easily because

`A(Avecx)=A^2vecx=0,` since we know that `A^2=0.`

Thus the column space is a subset of the null space, and since the column space is easily shown to be a vector space, the result follows.