Product of vector transpose and matrix

Question

Let $A$ be a $m\times n$ matrix and $\boldsymbol{v}$ be a $n$-dimensional vector and $\boldsymbol{u}$ be a $m$-dimensional vector. I have difficulty understanding $\boldsymbol{u}\cdot (A\boldsymbol{v})=\boldsymbol{u}^TA\boldsymbol{v}$, especially $\boldsymbol{u}^TA$. Based on my understanding $A\boldsymbol{v}$ returns a $m$-dimensional vector. I am unsure what kind object is $\boldsymbol{u}^TA$. I suspect it must be a $n$-dimensional vector, right? How the multiplication is defined in in $\boldsymbol{u}^TA$?