(n-1)-dimensional subspace is the null space of a linear functional

[yifli]

Given that N is an (n-1)-dimensional subspace of an n-dimensional vector space V, show that N is the null space of a linear functional.

My thoughts:

suppose \alpha_i(1\leq i \leq n-1) is the basis of N, the linear functional in question has to satisfy f(\alpha_i)=0.

Am I correct?

Thanks

 

[micromass]

Yes, that is already correct!

 

[HallsofIvy]

Given a basis for any vector space, any linear functional depends only upon its "action" on those basis vectors- that, is given f(v1)= x1, f(v2)= x2, ..., f(vn)= xn where the v's are the basis vectors and the a's are scalars, then any vector v can be written as v= a1v1+ a2v2+ ...+ anvn for some scalars, a1, a2, ..., an. Since f is linear, f(v)= a1f(v1)+ a2f(v2)+ ...+ anf(vn)=a1x1+ a2x2+ ... anxn.

Given any subspace U of vector space V, select a basis {u1, u2,...,um} for U and extend it to a basis for V. Define f(u1)= f(u2)= ...= f(um)= 0 and define f to be whatever nonzero values you like for the other basis vectors. Then the nullspace of f is exactly U.