Question

Let A be an m × n matrix such that ker(A) = {⃗0} and let ⃗v1,⃗v2,...,⃗vq be linearly independent vectors in Rn. Show that A⃗v1, A⃗v2, . . . , A⃗vq are linearly independent vectors in Rm.

Answer 1

Let us assume that Av₁, Av₂, . . . , Av_q are linearly dependent vectors in R^m. Then there exist scalars c₁,c₂,…,c_q, not all zero such that c₁Av₁ + c₂Av₂+…+ c_qAv_q = 0 or, A(c₁ v₁ +c₂ v₂+…+c_q v_q) = 0.

Now, since ker(A) = {0}, this implies that c₁ v₁ +c₂ v₂+…+c_q v_q = 0. However, since the vectors v₁,v₂,...,v_q are linearly independent in Rⁿ, hence c₁ = c₂ = …c_q = 0. This is a contradiction . Therefore, Av₁, Av₂, . . . , Av_q are linearly independent vectors in R^m.