# How is the dual space generated from the linear forms?

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).

• Well,, it is the space of all $1\times n$ (row) matrices. Jan 30 '21 at 0:37
• Any thoughts on the comments and answers you've received, Minsky? Jan 31 '21 at 12:02
• "Waiting" won't do you any good, if no one knows what you are waiting for. Failure to engage with other users just convinces people that you have lost interest in the question. Write comments! Let us know what you need, what's unsatisfactory with what you've got! Jan 31 '21 at 21:30
• Also, I'm not sure what you mean by "write an example". An example of what? Doesn't my answer give an example of how a linear functional ($T_w:V\to{\bf R}$) can be thought of as a vector, in that the collection of all the functionals $T_w$ (as $w$ goes through $V$) satisfies the definition of a vector space? Feb 1 '21 at 2:39
• OK, I've expanded my answer. Please don't tell me you can't understand it – tell me exactly what you don't understand, what sentence, what phrase, what word. Or maybe start by telling me what you do understand; vector space? linear transformation? linear functional? Feb 2 '21 at 4:42

## 2 Answers

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.

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 $$w^T\cdot v$$.