Let $V,W$ be finite-dimensional vector spaces over a field $F$.
Let $H:V\times W \rightarrow F$ be a bilinear map and $\alpha=\{v_1,...,v_n\} , \beta=\{u_1,...,u_m\}$ be ordered bases for $V,W$ respectively.
Then there exists a unique $n\times m$ matrix $A$ such that $A_{ij}=H(v_i,u_j)$.
"We call $A$ the matrix representation of $H$ with respect to $\alpha,\beta$ where $\alpha,\beta$ are ordered bases for $V,W$ respectively."
Is there any notation for this matrix $A$?
I saw that some write this matrix as $[H]_\beta$ if $\alpha=\beta$ (so that $V=W$), but i don't know what notation would be used when $V\neq W$.
If there does not exist a such being widely used notation, i will use $[H]_{\alpha\times \beta}$ for this matrix.
Is it ok?
I mean, is there any field of mathematics which uses a notation $[]_{\alpha\times\beta}$ so that my notation would cause some confusion?