The matrix representation for the basis of the vector space of the linear transformations $\mathcal{L}(V,\,W)$.

Question

While studying the linear algebra, I found out that the linear transformation $E_{ij}$ defined as follows provides the basis of the vector space of linear transformations from $V$ to $W$, or $\mathcal{L}(V,\,W)$. $$E_{ij}\left(\sum_{0\leq k<n}\alpha_{k}v_{k}\right)=\alpha_{j}v'_{i}.$$ Here, the basis for $V$ is $\mathcal{B}=\{v_{0},\cdots,v_{n-1}\}$ and the basis for $V'$ is $\mathcal{B'}=\{v'_{0},\cdots,v'_{m-1}\}$. Also, $0\leq j<n$, $0\leq i<m$, and $\alpha_{k}\in\mathbb{C}$. I want to know how to find out the matrix representation for $E_{ij}$. I know that the definition of the matrix representation for the linear map $\phi:V\rightarrow W$ is given by $\phi(v_{j})=\sum_{0\leq i<n}A_{ij}w_{i}$ but exactly what should I do to figure out the components of $A_{ij}$ in this case?

Answer 1

Note that $$E_{kl}(v_j) = E_{kl} \bigg( \sum_{0 \leq \mu < n} \delta_{j\mu}v_\mu \bigg) = \delta_{jl}v_k' = \sum_{0 \leq i < n} \delta_{ik}\delta_{jl}v_i'$$ and that $$\delta_{ik}\delta_{jl} = \begin{cases} 1 & \textrm{if } (i,j) = (k,l), \\ 0 & \textrm{if } (i,j) \neq (k,l). \end{cases}$$ That is, the matrix of $E_{kl}$ is the matrix that has a $1$ in the $(k,l)$-th entry and $0$ elsewhere.