Finding set of solutions to $AX=0$ for triangular matrix $A$.

Question

Let $\ A = (a_{ij})_{i,j=1}^n$ be a triangular n × n-matrix, that is, $\ a_{i,j}=0$ if $1 ≤ j < i ≤ n$.
Assume none of diagonal elements is equal to zero.
Find the set of all solutions $X\in\Bbb{R}^n$ of equations $AX = 0$.

I think I figured out what the matrix looks like, is the solution just the transpose? If so, how does one prove that?

Answer 1

Because $A$ is triangular, its determinant is the product of its diagonal entries. Because these are all non-zero, the determinant is non-zero. This means $A$ is invertible, and so $AX=0$ implies $X=0$.

Answer 2

Hint: You know the last component of X has to equal zero (why?). So what does that imply?