I need help with proving T^v is surjective. I have the following so far. How should I prove duality linear transformation?

Let V/K be a finite-dimensional vector space, and let 7 : V - V be an injective linear transformation. Let 7" : VV - V be the dual linear trans- formation. Prove that 7' is surjective. Proof. Since V/ K is a finite-dimensional vector space then let dim (V ) = n. Since 7 is surjective, then by the propsition from class T is surjective iff Im T = V. Then dim ( Im 7) = dim(V ) = n. From class notes, we know that dim(ker 7 ) + dim (Im 7) = dim(V). Hence, dim (ker 7) + dim(Im 7) = dim(V) => dim(ker T)+n = n => dim (ker T) = ker T = 02 Therefore, T is injective