TITLE: Verifying the examples of Dual Space
QUESTION [6 upvotes]: I am new to the concept of Dual space. Can someone please explain which of these are dual spaces and why? Also if $y$ is a polynomial on $x$, then what does $x(0)$ means?
• For $x \in P$, $y(x) = x(0)$
• Any vector space V, $y(x) = 0$; for every $x \in V $

REPLY [11 votes]: As Pedro said in his answer, the dual space of a vector space $V$ is a set whose elements are linear functions from $V$ to what Pedro called the "underlying field".  If this is unfamiliar to you, that probably means this mysterious "underlying field" is $\mathbb{R}$, the real numbers. So the dual space (sometimes denoted $V^*$) is the set of all linear functionals, i.e. all linear maps $$T: V \to \mathbb{R}$$
In other words, once you pick a basis for $V$, the dual space is just all functions represented by a row matrix $$(a_1 \ a_2 \ \dots \ a_n)$$
For example, if $V$ is the set of all $2 \times 2$ matrices, then the trace map $Tr: V \to \mathbb{R}$ is a linear functional (ie a linear map from $V$ to $\mathbb{R}$), since $$Tr(xA + yB) = x \ Tr(A) + y \ Tr(B)$$ where $x$ and $y$ are real numbers ("scalars") and $A$ and $B$ are matrices.  Mind you, this is not the entire dual space! This is just one element of the dual space. So $Tr \in V^*$. If you pick the basis $$\{ 
\begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}
\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}
\begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix}
\begin{pmatrix}0 & 0 \\ 0 & 1 \end{pmatrix}\}$$
for $V$, then the map $Tr$ is given by the row matrix $$(1 \ 0 \ 0 \ 1)$$
Footnote: The reason for saying "underlying field" is that linear algebra can be done over many different "objects": the rational numbers, complex numbers, and finite fields - integers modulo a prime number $p$, and oftentimes even the integers or other "rings".  But if that's not a point of interest now, then that's fine.