Problem : if $T: \Bbb{R}^3\to \Bbb{R}^4$ is one-to-one, describe the possible echelon forms of the standard matrix for a linear transformation $T$.
My solution is below.
To be one-to-one every column vectors must be independent, it means every columns must have pivot position. So I can form like this. $$ \begin{pmatrix} a&*&*\\ 0&b&*\\ 0&0&c\\ 0&0&0 \end{pmatrix} $$
Am I right? If not, how can I approach this problem?