What is the "role" of linear functionals in linear algebra?

Question

First, I apologize if my question is meaningless as it's often the case with this kind of question. I'll try to describe what bothers me in the best level of detail I can and I completely understand if there turns out to be no "answer". I frequently feel the need for a "narrative" of why the concepts were organized/constructed the way they are.

I've done a basic course in linear algebra - as it is customary - it was focused on computations and now I am doing the more mature course in linear algebra. I got a bit lost when I read about linear functionals. Precisely, why do we need them and why do the authors frequently say they are very important?

My guess is the following: We are trying to write linear algebraic ideas completely in terms of linear transformations. I guess this is the case because the vectors are defined as:

$$\textbf{x}=\sum_i a_i(\textbf{x}) \textbf{x}_i$$

Where $a_i(\textbf{x})$ are linear functionals, instead of:

$$\textbf{x}=\sum_i a_i \textbf{x}_i$$

Where $a_i$ are "merely" elements of the base field.

My new doubt is (supposing my guess is correct): What do we gain with this? They seem equivalent to me but it seems there is a description of the interaction between the two vector spaces $\Bbb{V}$ and $\Bbb{V}^*$ and also a description of what happens on $\Bbb{V}^*$ when we apply a linear transformation on $\Bbb{V}$. Why couldn't we go along with the second interpretation I gave? Perhaps, what we "gain" is that there is a simple description when we make changes between two radically different vector spaces and these changes are more easily "trackable" when everything is written in terms of linear transformations?

Answer 1

Your guess is a large part of the reason; algebraists prefer maps to elements. See the discussion of duals and compact categories in the Rosetta stone.

On a more concrete level, if you work over a commutative ring (rather than a field), the statements

• "each vector $\mathbf{x}$ can be written as $\mathbf{x} = \sum_i a_i\left(\mathbf{x}\right) \mathbf{x}_i$, where the $a_i$ are linear maps to the base ring, independent of $\mathbf{x}$"

and

• "each vector $\mathbf{x}$ can be written as $\mathbf{x} = \sum_i a_i \mathbf{x}_i$, where the $a_i$ are scalars that depend on $\mathbf{x}$"

(where, in both cases, $I$ is a fixed finite set, and the $\mathbf{x}_i$ are fixed vectors independent of $\mathbf{x}$) are not equivalent. The former statement holds if and only if your module is projective finitely generated (with $\left(\mathbf{x}_i\right)$ and $\left(a_i\right)$ being "dual generating systems"), whereas the latter holds if and only if your module is finitely generated. So requiring the coefficients $a_i$ to follow a global linear pattern, versus allowing them to arbitrarily depend on $\mathbf{x}$, makes a difference.