Here is the problem:
Let V be a finite-dimensional real inner product space, and let w $\in$ V be a fixed vector. Find the adjoint of T.
I have proved that the map T: V$\to$$R$ defined by T(v)=$<w,v>$ is a linear transformation.
Could anyone help me with the following steps?
Thanks!
Define $S: \mathbb R \to V$ by $\lambda \mapsto \lambda w$ and compute:
$$\langle S\lambda,v \rangle = \langle \lambda w,v \rangle = \lambda \langle w,v \rangle = \lambda \cdot T(v)$$
Hence - by definition - $S$ is the adjoint of $T$.
(Note that the inner product on $\mathbb R$ is just multiplication, i.e. $\lambda \cdot T(v)$ is actually $\langle \lambda, T(v) \rangle$)
Morally we have the following: $T$ is given by the row vector $w^T=(w_1, w_2, \dotsc, w_n)$, and its adjoint is given by the transposed matrix, i.e. the column vector $w=\begin{pmatrix}w_1\\w_2 \\ \vdots \\ w_n\end{pmatrix}$.