Onto and one-to-one linear transformations My book seems a bit unclear about onto and one to one linear transformations. Here is the excerpt:

and this is what I understand to be the rank of T:

rank(T) = dimension of column space of the original matrix.
Is the rank of T not the same thing as the dimension of W? Are these not the same thing? Can someone give me an example?

Yes of course for any matrix the dimension of $\operatorname{Col}(A)$ is equal the dimension of $\operatorname{Row}(A)$ and that number is by definition $\operatorname{rank}(A)$.
Therefore if $\operatorname{rank}(A)$ is equal to the dimension of $W$ it means that the set "columns of $A$" contains a basis for $W$ and therefore the transformaion is onto.
