I have a system of $L$ linear equations with $N$ unknowns ($N<L$). The equations themselves are not given, but I know that, when I write the equations in matrix format $A x = b$, all $N$ columns in $A$ are linearly independent (I cannot say the same for all rows). I am trying to show that if a solution exists, this will be unique. But I think that this cannot be shown without assuming that the L rows also are linearly independent.
My initial direction was to say that the rank of the matrix is $N$ (since $N < L$, and the columns are linearly independent), and to show that this will equal to the rank of the augmented matrix if there exists a solution. But if there exists rows that are not linearly independent, then the row rank of the matrix can be less than $N$, in which case we will have an underdetermined system of equations —> not necessarily a unique solution.
Can I somehow infer that all rows are linearly independent, if the columns are linearly independent? Alternatively, is there a more direct approach to show that if there exists a solution in this system of equations, this solution is unique.