# triangular matrix and linearly independence

Let $T$ be a triangular matrix where $t_{ii}\ne 0$ for all $i$. Show that the rows and the columns of $T$ are linearly independent. I think it is obvious from the structure of $T$. But I do not know how to prove.

• Note that $T$ and its transpose are invertible. – Julien Jun 14 '13 at 15:14

Consider the lower triangular case and let's deal with the row case. If $\sum_{i=1}^{n} \lambda_{i} r_{i} = 0,$ and not all $\lambda_{i}$ are $0,$ then choose $j$ as large as possible so that $\lambda_{j} \neq 0.$ Then $\lambda_{j}r_{j} = -\sum_{i=1}^{j-1} \lambda_{i}r_{i}.$ The right hand expression has $(j,j)$-entry $0,$ but the left hand side doesn't, a contradiction. (Strictly speaking I should have also said that there is more than one non-zero $\lambda_{i}$, which there is).