Suppose that L V rightarrow W is a linear transformation Sho

Suppose that L: V rightarrow 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 L is injective and let S V be linearly independent.

We will show that L(S) = {L(v) : v S} is linearly independent. So let a1L(v1) + · · · + anL(vn) = ~0 .

This implies that L(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 L(S) is linearly independent. Conversely suppose that L maps linearly independent sets to linearly independent sets and let v N(L). If v 6= ~0 then {v} is linearly independent, so {L(v)} is linearly independent. But if L(v) = ~0 this is impossible, since {~0} is linearly dependent. Thus v 6= ~0 and N(L) = {~0}, implying L is injective.