How is the dual space generated from the linear forms?

Question

Been reading about dual spaces without knowing much about it (not a mathematician). Now, after stumbled upon linear forms (function mapping vector space to number), that Wikipedia page reads:

The set of all linear functionals from $\mathbf{V}$ to $k$ , denoted by $\mathrm{hom}(V,k)$ is itself a vector space over $k$ with the operations of addition and scalar multiplication defined pointwise. This space is called the dual space of $\mathrm{V}$, or sometimes the algebraic dual space, to distinguish it from the continuous dual space.

The linear functional it talks about it's just $$ \begin{align} f(\vec{x}) &= \vec{w}\cdot{}\vec{x}\\ &= a_1 x_1 + a_2 x_2 + \ldots + a_n x_n \end{align} $$

how does this produce a space of vectors?

As I said before, not a mathematician so don't go over a million definitions (please).

Answer 1

Given an element $w$ of a vector space $V$, there is a linear map $T_w:V\to{\bf R}$ given by $T_w(v)=w\cdot v$. As $w$ runs through $V$, the set of these linear maps takes on the structure of a vector space, and it's called the dual space of $V$.

I'm now going to rewrite this in words of one syllable, as it were.

For each $w=(a_1,a_2,\dots,a_n)$ in ${\bf R}^n$ we can define a function $f_w$ from ${\bf R}^n$ to the real numbers by $f_w(x)=w\cdot x=a_1x_1+a_2x_2+\cdots+a_nx_n$ (where $x=(x_1,x_2,\dots,x_n)$).

If we take two of these functions, say, $f_w$ and $f_z$, we can add them to get a new function from ${\bf R}^n$ to the real numbers; if $g=f_w+f_z$ then $g(x)=f_w(x)+f_z(x)=w\cdot x+z\cdot x=(w+z)\cdot x=f_{w+z}(x)$. That is, it turns out that when you add two of these functions, you get another one of these functions, as $f_w+f_z=f_{w+z}$.

If we take one of these functions, $f_w$, and multiply it by a real number, we get a new function from ${\bf R}^n$ to the real numbers; if we call this new function $h=cf_w$, then $h(x)=cf_w(x)=c(w\cdot x)=(cw)\cdot x=f_{cw}(x)$. So, if you multiply one of these functions by a real number, you get another one of these functions, $cf_w=f_{cw}$.

So now if we consider the collection of all of these functions $f_w$, we have a set of functions with the properties that if you add two functions that are in the set, you get another function that's in the set, and if you multiply a function that's in the set by a real number, you get another function that's in the set. Now that's the beginning of the definition of a vector space – a vector space is a set where you can do addition, and where you can do multiplication by a real number, and where moreover these two operations have a whole bunch of other properties, like associativity and commutativity and zero element and additive inverses and distributive laws. And one can show that the collection of all these functions $f_w$ has all of these properties, so the collection is a vector space.

Answer 2

A particular linear function $f:V\to k$ is not producing a 'space of vectors', instead it is an element of a vector space, namely of the dual space of $V$.

$\hom(V,k)$ is the set of all linear functionals, and addition and scalar multiplication are done pointwise, and will satisfy the vector space axioms.

Additionally, if $V$ is finite dimensional with a basis $e_1,\dots,e_n$, then elements of $V$ are usually represented by column vectors (of the coordinates with respect the basis), and any linear map $f:V\to k$ can be represented by the row vector $w=(f(e_1),\dots,f(e_n))$, so that $f(v)=w\cdot v$ for all $v$ (coordinated in $e_i$), where $\cdot$ is actually matrix product, but it's the same as the dot product $w^T\cdot v$.