Suppose that L V W is a linear transformation Show that L i

Suppose that L : V W is a linear transformation. Show that L is one-to-one if and only if it maps every linearly independent set into a linearly independent set.

Solution

Assume that T is injective and let S V be linearly independent. We will show that T(S) = {T(v) : v S} is linearly independent. So let a1T(v1) + · · · + anT(vn) = ~0 . This implies that T(a1v1 +· · ·+anvn) = ~0, implying that a1v1 +· · ·+anvn = ~0 by injectivity. But this is a linear combination of vectors in S, a linearly independent set, giving ai = 0 for all i. Thus T(S) is linearly independent. Conversely suppose that T maps linearly independent sets to linearly independent sets and let v N(T). If v 6= ~0 then {v} is linearly independent, so {T(v)} is linearly independent. But if T(v) = ~0 this is impossible, since {~0} is linearly dependent. Thus v 6= ~0 and N(T) = {~0}, implying T is injective.