(i) Define the transpose of a matrix. If $V$ and $W$ are finite-dimensional real vector spaces, define the dual of a linear map $T: V \rightarrow W$. How are these two notions related?

Now suppose $V$ and $W$ are finite-dimensional inner product spaces. Use the inner product on $V$ to define a linear map $V \rightarrow V^{*}$ and show that it is an isomorphism. Define the adjoint of a linear map $T: V \rightarrow W$. How are the adjoint of $T$ and its dual related? If $A$ is a matrix representing $T$, under what conditions is the adjoint of $T$ represented by the transpose of $A$ ?

(ii) Let $V=C[0,1]$ be the vector space of continuous real-valued functions on $[0,1]$, equipped with the inner product

$$\langle f, g\rangle=\int_{0}^{1} f(t) g(t) d t$$

Let $T: V \rightarrow V$ be the linear map

$$T f(t)=\int_{0}^{t} f(s) d s$$

What is the adjoint of $T ?$