# 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!
If $\sigma = (e_1, \ldots, e_n)$ is the standard basis of $\mathbf R^n$ and $(1)$ is the standard basis of $\mathbf R$, then $Tv = [Tv]^{(1)} = [T]_\sigma^{(1)} [v]^\sigma = [T]_\sigma^{(1)} \cdot [v]^\sigma = [v]^\sigma \cdot [T]_\sigma^{(1)} = v \cdot [T]_\sigma^{(1)}$.
So in the case $V = \mathbf R^n$ we can understand the Riesz representation theorem as saying something we already knew. What Riesz representation theorem does is extend this same sort of "representability" to all finite-dimensional inner product spaces V and all linear functionals $T : V \to \mathbf F$.
this video talks about this for $\mathbf R^2$