In linear algebra, I've always thought about the notion of basis as something related to vector spaces, i.e., given a (real) vector space $V$, a basis is a (minimal) set of linear independent vectors $\mathcal{B} = \{\textbf{v}_1, \dots , \textbf{v}_n\}$ such that $V = Span(\mathcal{B})$. I was trying to understand what does exactly mean when people write the term basis when talking about linear maps. Let's say we fix two vector spaces $V$, $W$ with bases $\mathcal{B}_V$ and $\mathcal{B}_W$. Now if we consider the collection of linear maps $\mathcal{L} = \{L:V \rightarrow W \}$, how a basis for $\mathcal{L}$ can be formally defined?
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3The linear maps between two vector spaces form a vector space themselves. A basis of that vector space is defined exactly the same way as any other basis: a maximal set of linearly independent vectors, where the vectors are linear maps. – Vercassivelaunos Dec 22 '22 at 17:21
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The definition of basis is the same: a linearly independent set which also spans. An example of a basis, induced by the bases on $V,W$, is the collection $\{T_{ij}: 1\leq i\leq n, 1\leq j\leq m\}$ where $T_{ij}:V\to W$ is the unique linear map such that for all $k\in\{1,\dots, n\}$, \begin{align} T_{ij}(v_k)&= \begin{cases} w_j&\text{if $k=i$}\\ 0 & \text{else} \end{cases} \end{align} It is a standard, but useful and very informative exercise for you to prove that this collection of maps actually forms a basis for the space $\text{Hom}(V,W)$ for all linear maps from $V$ into $W$.
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