Dual Vector Spaces and Maps

What s the dual space V^* of a vector space V?

The dual space of V, denoted V^*, is the set of all linear functionals from V to \mathbb{F}: V^*=\{f:V\rightarrow\mathbb{F}|f \ is \ linear\}. It is itself a vector space over \mathbb{F} with dimension equal to dim(V).

What is a linear functional?

A linear functional is a linear map f:V\rightarrow\mathbb{F}. It assigns a scalar to each vector and satisfies f(\alpha v +\beta w)=\alpha f(v)+\beta f(w) for all v,w\in V and \alpha,\beta\in\mathbb{F}.

What is the dual basis of a basis B=\{v_1,...,v_n\}?

The dual basis B^*=\{f_1,…,f_n\}\subseteq V^* satisfies f_i(v_j)=\delta_{ij} (1 if i=j, 0 otherwise) It “reads off” coordinates in the original basis.

Existence and Uniqueness of dual basis Theorem

Every basis \{v_1…,v_n\} of V has a unique dual basis \{f_1,…,f_n\} in V^* such that f_i(v_j)=\delta_{ij}.

What is the evaluation map ev_v:V^*\rightarrow\mathbb{F}?

Given v\in V, the map ev_v(f)=f(v) assigns to each linear functional its value on v. This gives a duality pairing between V and V^*.

What is the double dual of a vector space V?

The dual of the dual space V^{**}=(V^*)^*. There is a natural isomorphism V\cong V^{**} identifying each v\in V with the evaluation map ev_v\in V^{**}.

What is the transpose (dual map) of a linear map T:V\rightarrow W?

The dual map (also called the transpose) is T^*:W^*\rightarrow V^*, T^*(g)=g\circ T. It “pulls back” functionals from W to V.

How do matrix representations of T and T^* relate?

If [T]=A then [T^*]=A^T. The matrix of the dual map is the transpose of the matrix of the original map (assuming dual bases are used).

Let T:\mathbb{R}²\rightarrow \mathbb{R}"², T(x,y)=(x+y,y). What is T^*?

Matrix of T: A=\left[\begin{array}{cc}1&1\\0&1\end{array}\right]\Rightarrow T^* has matrix A^T=\left[\begin{array}{cc}1&0\\1&1\end{array}\right]. T^* acts on \mathbb{R}²\rightarrow\mathbb{R} functionals by composition.