I have the transformation from $\mathbb R^2$ to $\mathbb R^3$. With the matrix $$ \begin{pmatrix} 1\quad 1 \\ 0\quad 1 \\ 2\quad 1 \end{pmatrix} $$ According to what people write on Stack overflow all I need is to reduce the matrix and then take the columns which have a leading number there. So here I gotta take both columns and get. $(102), (111)$ being the basis and $\dim(\operatorname{Im})=2$.
Point is, I read a book now. They do it in a different way. They reduce the matrix by adding a third column of $a,b,c$. And then when reducing they find the image. Then they get the basis equals $(102) , (011)$.
These are different results. Maybe their span provides the same plane?? I am not sure. That's why I am asking. They have same dim but what tells me they are the same thing? When I compare the two they don't seem at all to be equal. I need help if someone can clarify what's going on here.