Question

Let A be an m x n matrix. Show that if A has linearly independent column vectors, then N(A) = {0}.

Answer 1

Let the column vectors of A be:v1,v2,…,vnIf the matrix A has linearly independent column vectors, then the only linear combination of the vectors that equals 0 is the zero combination. In other words, if:c1v1+c2v2+⋯+Cn.Vn=0⟹c1=c2=⋯=Cn=0However:c1v1+c2v2+⋯+Cn.vn=0⟺Ax=0where x is the vectorx=(C1,C2,…,Cn)and if Ax=0, this means that x is in the null space of A. Since the only vector that makes that linear combination 0 is the 0 vector, it follows that the 0 vector is the only vector in the null space.