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I previously asked a question about The notation $x\in\mathsf{F}^n$ means $x$ a tuple or a column vector?. After re-read all those related chapters, my conclusion, which all words can be found in the book, is that any element $x\in\mathsf{F}^n$ is a tuple, which can be written as two forms: vertically a column vector or as horizontally a row vector. But this didn't solved the confusion I really want to know.

My current understanding is that: A linear transformation is an abstract idea. To concretize it into one of its matrix-forms I need the idea of ordered basis s.t. it can be seen as a matrix relative to the two bases. But it's confusing that a left-multiplication transformation is defined using a matrix and need the idea of matrix multiplication. Say $$\large L_A:\mathsf{F}^n\to \mathsf{F}^m, L_A(x)=Ax,\textrm{for any}\ x\in\mathsf{F}^n,$$

which is a left-multiplication transformation defined using a matrix $A$. Since I haven't decided which are the two bases to be used to represent those $x\in\mathsf{F}^n,L(x)\in\mathsf{F}^m,$ how to even compute the multiplication $Ax$? It seems like there is implicitly a by default we use the standard ordered bases of $\mathsf{F}^n$ and $\mathsf{F}^m$ before doing the multiplication. So which part did I miss?

linear_combinatori_probabi
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    Regarding your original confusion, the abstract and the concrete are often related. In this case, abstract linear transformations between abstract vector spaces over $F$ are related to matrix transformations between coordinate vector spaces $F^n$, the relation being governed by choices of bases. – Lee Mosher Apr 15 '18 at 13:24

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When one uses the notation $F^n$ for the vector space of $n$-tuples of elements of $F$, there is indeed an implicit standard ordered basis $e_1,...,e_n$, where $e_i$ has a $1$ in the $i^{\text{th}}$ position and $0$'s elsewhere.

In the displayed formula, the matrix $A$ is given first, and then it is used to define the linear transformaion $L_A : F^n \to F^m$ (so you can think of this as a function which inputs the matrix $A$ and outputs the linear transformation $L_A$). This works by using the implicit standard ordered bases of $F^n$ and $F^m$.

But this would not work for defining an abstract linear transformation $L : V \to W$ between abstract vector spaces $V$ and $W$ over $F$, unless bases for $V$ and $W$ were given or were specified in some manner.

Lee Mosher
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