Snippet taken from "Linear Algebra" by Jim Hefferon (4th edition):
1.2 Definition Suppose that $V$ and $W$ are vector spaces of dimensions $n$ and $m$ with bases $\mathrm{B}$ and $\mathrm{D},$ and that $\mathrm{h}: \mathrm{V} \rightarrow \mathrm{W}$ is a linear map. If $$ \operatorname{Rep}_{\mathrm{D}}\left(\mathrm{h}\left(\vec{\beta}_{1}\right)\right)=\left(\begin{array}{c} \mathrm{h}_{1,1} \\ \mathrm{~h}_{2,1} \\ \vdots \\ \mathrm{h}_{\mathrm{m}, 1} \end{array}\right) \quad \ldots \quad \operatorname{Rep}_{\mathrm{D}}\left(\mathrm{h}\left(\vec{\beta}_{\mathrm{n}}\right)\right)=\left(\begin{array}{c} \mathrm{h}_{1, \mathrm{n}} \\ \mathrm{h}_{2, \mathrm{n}} \\ \vdots \\ \mathrm{h}_{\mathrm{m}, \mathrm{n}} \end{array}\right)_{\mathrm{D}} $$ then $$ \operatorname{Rep}_{\mathrm{B}, \mathrm{D}}(\mathrm{h})=\left(\begin{array}{cccc} \mathrm{h}_{1,1} & \mathrm{~h}_{1,2} & \ldots & \mathrm{h}_{1, \mathrm{n}} \\ \mathrm{h}_{2,1} & \mathrm{~h}_{2,2} & \ldots & \mathrm{h}_{2, \mathrm{n}} \\ & \vdots \\ \mathrm{h}_{\mathrm{m}, 1} & \mathrm{~h}_{\mathrm{m}, 2} & \ldots & \mathrm{h}_{\mathrm{m}, \mathrm{n}} \end{array}\right)_{\mathrm{B}, \mathrm{D}} $$ is the matrix representation of h with respect to B, D.
Here $h$ is homomorphoism. What I am not able to understand that if there is a linear mapping, $f:V\to W$, then according to a previous theorem in the same book, the bases of $V$ maps to bases of $W$. Therefore, if $B$ is the bases of $V$ containing $n$ vectors $\langle B_1, B_2, \dots, B_n\rangle$, then $\langle h(B_1), h(B_2), \dots, h(B_n)\rangle$ is the bases for $W$.
Now shouldn't in the definition of the attached screenshot $h_ij = 1$; if $i=j$ and $h_ij = 0$; if $i\neq j$? So that the final matrix have to be an identity matrix? What's going on?
(The book is available from https://hefferon.net/linearalgebra/ and this definition is on page 214. The notation $\mathrm{Rep}_B(\vec v)$ is defined on page 124.)
You've not told us how this notation is defined in the book.
I take it that $B=(\beta_1,\dots, \beta_n)$ is an ordered basis of $V$ and $D=(\delta_1,\dots,\delta_m)$ is an ordered basis of $W$.
I think that if you look at the definition of $\text{Rep}_D (v)$ you will see that the hypothesis is that for each $i$ the vector $h(\beta_i)$ can be written $$ h(\beta_i)=h_{1,i}\delta_1+\dots +h_{m,i}\delta_m . (*) $$
If you now look at the definition of $\text{Rep}_{B,D}(f)$ you will see that $(*)$ are exactly the equations you need to make $\text{Rep}_{B,D}(h)$ exactly what the book says.