Adjoint Linear Algebra I am trying to understand the definition of an adjoint in linear algebra but there is one step in the definition that I can't get. 
Definition: Assume T is a linear operator on a finite dimensional inner product space V. Then an adjoint $T^*$ exists such that $<x,T^*y> = <Tx,y>$.
So we start with $<T^*x,y>$ which equals $\overline{<y,T^*x>}$ by definition of the inner product. The proof then says that $\overline{<y,T^*x>}$  must equal $\overline{<Ty,x>}$ which I don't understand. The change of $T^*x$ to $x$ is from the definition of the adjoint but why must Ty = y? It doesn't say that T must be the identity transformation.

You aren't having $T^*y=y$, you are moving a $T^*$ from the right side and turning it into a $T$ on the left.  The definition of adjoint is that 
$$\langle x, T^* y \rangle = \langle Tx, y \rangle$$
regardless of what $x$ and $y$ are. Even if $x$ and $y$ are swapped.
