[tex]T: R^n \rightarrow R^m[/tex] is a linear transformation.
a) Calculate Dim(ran(T)) if T is one to one.
b) Calculate Dim(ker(T)) if T is onto.
The Attempt at a Solution
a) I need to calculate the dimension of range of T if it's 1-1.
So there is a property that: ran(T) = col(A) (where A is the standard matrix). And hence Dim(ran(T)) = Dim(col(T)). Since T is 1-1 Ax=0 has the trivial solution.
I'm not sure if I'm on the right track & I don't know where to go from here...
b) I need to "calculate" the dimension of kernel of T if it's onto.
I know that Dim(ker(T)) = Dim(null(A)) and of course Dim(null(A)) (that is the dimension of the null space of standard matrix A) is the nullity(A).
If the linear transformation T is onto then the linear system Ax=b must be consistent for every b in Rn.
Again I'm stuck here but this is my attempt so far...