Linear Transformations, Span, and Independence

[tandoorichicken]

Is there a linear algebra theorem or fact that says something like

For a linear transformation T:Rn -> Rm and its standard m x n matrix A:
(a) If the columns of A span Rn the transformation is onto.
(b) If the columns of A are linearly independent the transformation is one-to-one.

Is this correct? I can't find it anywhere in my textbook but it may have been mentioned in lecture. Any insight would be appreciated.

 

[Galileo]

If m>n, then can T be onto? If the colomns are independent, T(Rn) will be an n-dimensional subspace of Rm.

b) is true, as you can easily prove yourself. Show that T(v1)=T(v2) implies that v1=v2.
Hint: If you column-vectors are independent you can use them as a basis.