Answer
$u^{T}v = {u}^{T}v$
$(u{v}^{T})^{T}= vu^{T}$
Work Step by Step
The inner product $\mathrm{u}^{T}v$ is a real number (a 1$\times$1 matrix).
Its transpose equals itself.
By Theorem 3, $(\mathrm{u}^{T}v)=v^{T}(u^{T})^{T}=v^{T}u$
so $\mathrm{u}^{T}v$ = $\mathrm{u}^{T}v.$
The outer product $uv^{T}$ is an $n\times n$ matrix. By Theorem 3,
$(\mathrm{u}\mathrm{v}^{T})^{T}=(\mathrm{v}^{T})^{T}\mathrm{u}^{T}= vu^{T}$
Both results are confirmed with the results of the previous exercise.
