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:\[v_1,v_2,\ldots,v_n\]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:\[c_1v_1+c_2v_2+\cdots+c_nv_n=0\Longrightarrow c_1=c_2=\cdots=c_n=0\]However:\[c_1v_1+c_2v_2+\cdots+c_nv_n=0\iff Ax=0\]where x is the vector\[x=(c_1,c_2,\ldots,c_n)\]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.