Matrix free linear map decomposition

Question

Consider $a_1,\dots,a_n\in\mathbb{R}^n$ and identify $a_j\in\mathcal{L}(\mathbb{R},\mathbb{R}^n)$ via $\varphi\mapsto \varphi1$.

Also, consider $A\in\mathcal{L}(\mathbb{R}^n)$ given by $$A\colon (x_1,\dots,x_n)\mapsto a_1x_1 + \dots + a_nx_n\tag{$\star$}$$

What's the name or symbol of the map $$\mathcal{L}(\mathbb{R},\mathbb{R}^n)\times\dots\times\mathcal{L}(\mathbb{R},\mathbb{R}^n)\to\mathcal{L}(\mathbb{R}^n,\mathbb{R}^n),\quad(a_1,\dots,a_n)\mapsto A$$ where $A$ and $a_1,\dots,a_n$ are related as in $(\star)$? I'd like to write e.g. $A=a_1\otimes\dots\otimes a_n$. Is there some higher level concept that induces that map? (E.g. some time ago I wondered if this map would be the tensor product).

The matrix equivalent would be saying that the columns of $A$ are the column vectors $a_1,\dots,a_n$ and writing $A=\begin{bmatrix}a_1& \dots &a_n\end{bmatrix}$. However, I'd like to keep things matrix free.

Thanks in advance.

Answer 1

If I understand your question correctly, you are looking for an n-linear functional such that $(x_1, . . ., x_n) \to \sum_{i=1}^{n}a_{i}x_i$. Note that the scalar product of two arbitrary vectors $u, v$ is the following: $$<u, v> = u_1v_1 + ... + u_nv_n = \sum_{i=1}^{n}u_iv_i$$

So, for the map that you mentioned, we could define $M$ as the row vector with elements being the column vectors $a_i$ so that the scalar product $<M, X> = a_1x_1 + ... + a_nx_n.$

In other words, you are just looking for the scalar product operator.

Answer 2

I am not sure I understand what you are asking, but maybe you are looking for the map from a vector space to its dual, which we could write as $a \mapsto \langle a, \cdot \rangle$. In general this depends on a metric $\langle \cdot,\cdot\rangle$ (or some other preferred non-degenerate bilinear form). In $R^n$ it is often implicitly assumed that you use the Euclidean one and, with respect to the canonical basis, what you are doing is, for $a\in R^n$ $$a\mapsto \alpha\in (R^n)^*=\mathcal{L}(R^n,R),$$ where, for any $v\in R^n$, $$\alpha (v) =a^T v.$$