Suppose T Rn Rm is a linear transformation Let v1 v2 vp

Suppose T : Rn Rm is a linear transformation. Let v1, v2, . . . , vp be a set of linearly independent vectors in Rn. Show that T (v1), T (v2), . . . , T (vp) is a set of linearly independent vectors in Rm.

Solution

Let S = {v₁,v₂, :::,v_p}

Let the set of images

T(S) = {T(v₁), T(v₂), :::; T(v_p)}

If S is linearly dependent, then there exist co- efficients c₁,c₂, ...,cp not all of them zero such that

c₁v₁ + c₂v₂ + ::: + c_pv_p = 0_V

Apply the lienar transformation T to both sides yields

c₁T(v₁) + c₂T(v₂) + ::: + c_pT(v_p) = 0_W

that is, there exist co-efficients c₁, c₂, ...,c_p not all of them zero such that the linear combination of the transformed vectors is 0_W. This means that S\'=T(S) is a set of linearly dependent vectors.