# Thread: show columns are linearly independent if homogeneous system has only trivial solution

1. ## show columns are linearly independent if homogeneous system has only trivial solution

Let A be an m x n matrix. Show that the columns of A are linearly independent iff the homogeneous system Ax=0 has just the trivial solution.

I think I need to use the fact that a homogeneous system of n linear equations in n unknowns has a nontrivial solution iff rank(A) < n

2. You only need to express the system $\displaystyle Ax=0$ in the form $\displaystyle x_1C_1+\ldots+x_nC_n=0$ and apply the definitions of linearly independence and solution of a system.