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

  1. Oct 6, 2012 #1
    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. jcsd
  3. Oct 6, 2012 #2
    Yes


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

    DonAntonio
     
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