i'd be extremely grateful if anyone can help me out with this

-----

Suppose V is a ﬁnite dimensional inner product space and T : V → V a linear operator.

1. Prove im(T∗ ) = (kerT )⊥.

2. Prove rank(T ) = rank(T∗ ).

----

T∗ should be displayed as T^*, as is (kerT )⊥ should be (kerT)^⊥