# Matrices for linear maps

Given a linear map $$g : F^n \to F^m$$ ,and standard bases of $$F^n$$ and $$F^m$$ denoted by $${e_{i}}$$ and $${e_{j}^{'}}$$ respectively, \begin{aligned} & g(e_{j})=\sum_{i=1}^m a_{i,j}(e_{i}^{'}) \\ \end{aligned} If v belongs to $$F^n$$ we have

\begin{aligned} & g(v)=g(\sum_{j=1}^n \beta_j (e_{j})) \\ &=\sum_{j=1}^n \beta_j g(e_{j})\\ &=\sum_{j=1}^n \beta_j \sum_{i=1}^m a_{i,j}(e_{i}^{'})\\ \end{aligned}

I don't understand how the next two steps follow from above : \begin{aligned} &=\sum_{i=1}^n(\sum_{j=1}^m a_{i,j}\beta_j )(e_{i}^{'})\\ & = Av \end{aligned}

How is the summation index change correct? And how are we getting matrix A and vector v from the second last step?

Edit :

Assuming the index change is a typo, what I understand of the second last step is $$a_{i,j}$$ are elements of matrix A, and for each column of A a certain $$\beta_j$$ is multiplied to the elements of the entire column, summed up and then multiplied to a basis vector $$e_{i}^{'}$$ which means we are getting a linear combination of basis vectors of $$F^m$$. But it is not clear to me how this is factored to A and v, where v is a linear combination of basis vectors of $$F^n$$

N.B. Taken from Matrix Theory book by David Lewis.

The index change seems to be a typo, correct would be \begin{aligned} g(v)&=g(\sum_{j=1}^n \beta_j (e_{j})) \\ &=\sum_{j=1}^n \beta_j g(e_{j})\\ &=\sum_{j=1}^n \beta_j \sum_{i=1}^m a_{i,j}(e_{i}^{'})\\ &=\sum_{i=1}^m(\sum_{j=1}^n a_{i,j}\beta_j )(e_{i}^{'})\\ & = Av \end{aligned} The last identity is just the definition of the matrix vector product, where $$A$$ consists of the entries $$a_{i,j}$$.
Edit: Actually the last identity is a slight abuse of notation, since $$v$$ is not an element of $$\mathbb R^m$$. The vector of the coefficients $$\vec \beta:=(\beta_j)_{j=1,\ldots,m}$$ is though. So $$Av$$ here means the linear combination of the basis vectors $$e'_j$$ with coefficients $$(A\vec\beta)_j$$ i.e. $$Av=\sum_{j=1}^m(A\vec \beta)_je'_j$$