User:IssaRice/Linear algebra/Riesz representation theorem

Let's take the case where $V=\mathbf {R} ^{n}$ and the inner product is the usual dot product. What does the Riesz representation theorem say in this case? It says that if we have a linear functional $T:\mathbf {R} ^{n}\to \mathbf {R}$ then we can write $T$ as $Tv=v\cdot u$ for some vector $u\in \mathbf {R} ^{n}$. But we already know (from the correspondence between matrices and linear transformations) that we can represent $T$ as a 1-by-n matrix. And v can be thought of as a n-by-1 matrix. And now the dot product is the same thing as the matrix multiplication!