Linear transformation problem

Hi! Have some troubles with solving one problem in my assignment, would be great if somebody could help me. The problem is following:

Let *n>0* and let *L*_{n,k }, whear *1<=k<=n* be a subspace spanned by vectors *e*_{j} for *j=1...k*. Show that it exist a linear transformation T:R^{n}=>R^{(n-k)} such as L_{n,k}=*Ker*(T)

Re: Linear transformation problem

You can always find a **basis** for contained in then extend it to a basis for . Define for , T(v)= 0 for v in the extended basis. T(v) is defined "by linearity" for all other v in . That is, since , we can write and then . Then Tv= 0 if and only if v is in the subspace.

Re: Linear transformation problem

Thanx for answe! Could you please spicify what do you mean by saying that Tv=0, is it a transformation of v? What is v if it can be represented as v=a(1)e(1)+....b(n-k)*f(n-k). And if you said that T(e_j)=0, how can Tv be equal to a_1+a_2+....+a_k if the all supposed to be transformed into null!?