- #1

- 1,271

- 12

## Homework Statement

[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 R

^{n}.

Again I'm stuck here but this is my attempt so far...