Dual of a Vector Space 

Definition

MotivationIn Mathematics, it is very much possible to translate concepts, structures, or statements into other concepts, mathematical structures, or statements. This translation of one mathematical structure to another is termed as 'duality', where the two structures are the duals of each other. For example, a cube and octahedron are very closely related. If we choose the centers of all the six faces of a cube, then those will be the vertices of an octahedron. And, then we can say that the octahedron is the 'dual' of the cube. Interestingly enough, the converse i.e., the centers of the eight triangular faces of the octahedron will be the vertices of a cube and therefore, a cube is the dual of an octahedron. It can also be easily observed that the number of edges of two duals is the same, and the number of vertices of one dual is the number of faces of the other dual. So, the principle of duality can be considered to be one of the illuminating way of constructing a dual of any regular polyhedra. 2 
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Figure 1: An Octahedron is the dual of a Cube 
Bird's Eye ViewThe dual of an object is of the same kind as the object itself. In the previous example, dual of the cube is an octahedron and both of them are polyhedrons. Similarly, if we consider a set S and let X be its subset such that X $\subseteq$ S, then the dual of subset X will also be a subset of the set S. Conversely, the dual of the subset X^{c} will be the subset X, since (X^{c})^{c }= X and hence, they are duals of each other. 4 
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Figure 2: The dual of a subset is its complement. 
Duality is a property of algebraic structures, holds that the two concepts or operations are interchangeable. 7 
Context of the DefinitionLet $V$ be a vector space over a field $F$, then the dual of the vector space $V^*$ is a vector space that contains all the functions $T: V \rightarrow F$, where $T$ is called a linear functional. A Linear Functional is a function that accepts a vector $v\in V$ as input and gives out an element of the field $F$ over which the vector space $V$ is defined. 1) $T(u + v)$ = $T(u)$ + $T(v)$, $\forall u, v \in V$
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Figure 3: Linear Functional on Vector Space 
Let's assume the field to be $\mathbb{R}$, meaning that the linear functional gives out a real number as an output. If you take all the possible ways in which the linear functional will take vectors as input and give a real number as an output, you obtain the dual vector space $V^*$. Consider $V$ = $\mathbb{R}^2$ and $T: \mathbb{R}^2 \rightarrow \mathbb{R}$, then $T(x, y)$ = $ax + by$ will be an element of the dual vector space $V^*$ $\forall a, b \in \mathbb{R}$ since, $ax + by$ = $k$ where $k$ is a real number. If $V$ = $\mathbb{R}^3$ and $T: \mathbb{R}^3 \rightarrow \mathbb{R}$, then $T(x, y, z)$ = $ax + by +cz$ will be an element of the dual vector space $V^*$.
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Dual Basis of a Vector SpaceGiven a basis $B = (v_1, v_2, . . . . . . . , v_n)$ of the vector space $V$, there exists a dual basis $B^*$ = (${T_1}, {T_2}, . . . . . , {T_n}$) of the dual space $V^*$, If i = j, then $\delta_{ij}$ =1 and if i $\neq$ j, then $\delta_{ij}$ = 0. 13 
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Figure 4: Basis of a vector space and its dual. 
Consider a vector space $V = \mathbb{R}^2$ whose basis is $B = \{(2,1) ,(3,1)\}$, then to find the dual basis $B^*$ the following Step 1: Let $B^*$ = $\{{T_1}, {T_2}\}$, by definition ${T_1}(v_1) = 1$ $\implies$ ${T_1}(2, 1) = 1$ and, Step 2: ${T_1}(2, 1) = 1$ $\implies {T_1}[ 2(1, 0) + 1(0, 1)]$ = 1 $\implies$ $2 {T_1}(1, 0) + 1{T_1}(0, 1) = 1$  (i) Step 3: On solving the equations (i) and (ii), we get; Step 4: ${T_1}(x, y)$ = $x{T_1}(1, 0) + y{T_1}(0, 1)$ $\implies$ $x(1) + y(3)$ = $ x + 3y$ $\implies {T_1}(x, y) = x + 3y$ Step 5: Similarly, by using ${T_2}(2, 1)$ = 0 and ${T_2}(3, 1)$ = 1, we get ${T_2}(1, 0)$ = 1 and ${T_2}(0, 1) = 2$ Therefore, the basis of the dual vector space $V^*$ is given by the equation of two straight lines. 15 
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Figure 5: Geometrical Illustration of the example above. 
ApplicationsLinear Programming is considered to be a very good application of duality in which there the relationship between a given linear program and the other related linear program is developed.
HistoryIt was in 1907 when the definition of a continuous linear operation on the space $L^{2}([a,b])$ was given by Riesz. this is, in slightly modernized notation, an operation which for any $f\in L^{2}$ gives a number $U(f)$ such that $U$ is a linear map and such that whenever $f_n \rightarrow f$ in $L^{2}$ we have $U(f_n) \rightarrow U(f)$. Then he shows that for each continuous linear operation $U$ there exists a function $k$ such that $U(f) = \int_{a}^{b} f(x) k(x) dx$ for all $f\in L^{2}$. 17 
Pause and Ponder1) Is it possible for a sphere to have a dual? 2) Can you think of a linear functional that will map all the real matrices to a real number? 18 
Reference[1] https://www.cis.upenn.edu/~cis515/cis51518sl6.pdf [2] https://mathworld.wolfram.com/DualVectorSpace.html [3] https://ekamperi.github.io/mathematics/2019/11/17/dualspacesanddualvectors.html#definition [5] https://sites.math.northwestern.edu/~scanez/courses/334/notes/dualspaces.pdf [6] Per Manne (https://math.stackexchange.com/users/33572/permanne), History of Dual Spaces and Linear Functionals, URL (version: 20120703): https://math.stackexchange.com/q/166118 [7] DisintegratingByParts (https://math.stackexchange.com/users/112478/disintegratingbyparts), Who was the first to use dual space?, URL (version: 20140305): https://math.stackexchange.com/q/700572 19 
Further Reading[1] https://en.wikipedia.org/wiki/Duality_(mathematics) [2] https://www.math.upenn.edu/~aaronsil/Math312Spring2013/DualSpaces.pdf 20 
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