User:IssaRice/Linear algebra/Dual basis

see p. 224 of https://terrytao.files.wordpress.com/2011/06/blog-book.pdf

the following example is based on p. 115 of https://terrytao.files.wordpress.com/2016/12/linear-algebra-notes.pdf

Let $V$ be the space of all mixtures of CO2 and CO, and let $\beta :=(C{O}_2,CO)$ and ${\beta }^{\prime }:=(C,O)$.

The change of coordinate matrix, from $\beta$ to ${\beta }^{\prime }$, is then $[I_V{]}_{\beta }^{{\beta }^{\prime }}=\left(\begin{array}{cc}1& 1\\ 2& 1\end{array}\right)$.

The change of coordinate matrix, from ${\beta }^{\prime }$ to $\beta$, is the inverse of $[I_V{]}_{\beta }^{{\beta }^{\prime }}$, and we have $[I_V{]}_{{\beta }^{\prime }}^{\beta }=([I_V{]}_{\beta }^{{\beta }^{\prime }}{)}^{-1}=\left(\begin{array}{cc}-1& 1\\ 2& -1\end{array}\right)$.

Now let us extend this example to discuss dual spaces.

Given some mixture (i.e. linear combination) $a\times C{O}_2+b\times CO$, the dual basis of $\beta$ consists of two linear functionals $({\phi }_1,{\phi }_2)$ such that

${\phi }_1(a\times C{O}_2+b\times CO)=a$ ${\phi }_2(a\times C{O}_2+b\times CO)=b$

Similarly, given some mixture $c\times C+d\times O$, the dual space of ${\beta }^{\prime }$ consists of two linear functionals $({\psi }_1,{\psi }_2)$ such that

${\phi }_1(c\times C+d\times O)=c$ ${\phi }_2(c\times C+d\times O)=d$