# Matrix corresponding to linear transformation is invertible iff it is onto?

1. Oct 6, 2012

### Aziza

Let A be a nxn matrix corresponding to a linear transformation.
Is it true that A is invertible iff A is onto? (ie, the image of A is the entire codomain of the transformation)
In other words, is it sufficient to show that A is onto so as to show that A is invertible?
That was what my professor said but I am having trouble understanding this..could someone please prove this or direct me to a proof?

2. Oct 6, 2012

### DonAntonio

Yes

$$A\,\,\text{is onto}\,\,\Longleftrightarrow \dim(Im A)=n\Longleftrightarrow \dim(\ker A)=0\Longleftrightarrow A\,\,\text{ is }\,\,1-1$$

DonAntonio