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

Let the column vectors of A be: v1, v2, ..., vn If 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: c1 v1 + c2 v2 + ⋯ + Cn. Vn = 0⟹ c1 = c2 = ⋯ = Cn = 0 However: c1 v1 + c2 v2 + ⋯ + Cn. vn = 0⟺Ax=0 where x is the vector x = (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.