Let v1vk be a basis for a subspace S of an ndimensional vect

Let {v1,...,vk} be a basis for a subspace S of an n-dimensional vector space V. Prove that there esists a linear mapping L:V right arrow V such that ker(L)=S.

Solution

Let fu1;u2; : : : ;upg be a set of vectors with p > n. Since any set of more than n vectors of Rn is
linearly dependent, the vectors [u1]B; [u2]B; : : : ; [up]B of Rn must be linearly dependent. Then there exist
constants c1; c2; : : : ; cp, not all zero, such that
c1[u1]B + c2[u2]B + .... + cp[up]B = 0:
Thus
[c1u1 + c2u2 + .........+ cpup]B = c1[u1]B + c2[u2]B + ...... + cp[up]B = 0 = [0]B:
Note that the coordinate transformation is one-to-one. It follows that
c1u1 + c2u2 + ....... + cpup = 0:
This means that the vectors u1;u2; : : : ;up are linearly dependent by de¯nition   get that there esists a linear mapping L:V right arrow V such that ker(L)=S.