User:IssaRice/Linear algebra/Dual basis: Difference between revisions

Revision as of 05:37, 30 July 2019

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 $β : = (C O_{2}, C O)$ and $β^{'} : = (C, O)$ .

The change of coordinate matrix, from $β$ to $β^{'}$ , is then $[I_{V}]_{β}^{β^{'}} = (\begin{matrix} 1 & 1 \\ 2 & 1 \end{matrix})$ .

The change of coordinate matrix, from $β^{'}$ to $β$ , is the inverse of $[I_{V}]_{β}^{β^{'}}$ , and we have $[I_{V}]_{β^{'}}^{β} = ([I_{V}]_{β}^{β^{'}})^{- 1} = (\begin{matrix} - 1 & 1 \\ 2 & - 1 \end{matrix})$ .

Now let us extend this example to discuss dual spaces.

Given some mixture (i.e. linear combination) $a \times C O_{2} + b \times C O$ , the dual basis of $β$ consists of two linear functionals $(φ_{1}, φ_{2})$ such that

$φ_{1} (a \times C O_{2} + b \times C O) = a$

$φ_{2} (a \times C O_{2} + b \times C O) = b$

Similarly, given some mixture $c \times C + d \times O$ , the dual space of $β^{'}$ consists of two linear functionals $(ψ_{1}, ψ_{2})$ such that

$ψ_{1} (c \times C + d \times O) = c$

$ψ_{2} (c \times C + d \times O) = d$

Now we can ask, given $φ_{1}$ , how can we write it in terms of $ψ_{1}, ψ_{2}$ ?

Given the mixture $a \times C O_{2} + b \times C O$ , we can write this as $(a + b) \times C + (2 a + b) \times O$ . Thus, $ψ_{1} (a \times C O_{2} + b \times C O) = a + b$ and $ψ_{2} (a \times C O_{2} + b \times C O) = 2 a + b$ . In other words, our task is to express $a$ in terms of $a + b$ and $2 a + b$ . We get $(2 a + b) - (a + b) = a$ , so $φ_{1} = - ψ_{1} + ψ_{2}$ .

Similarly, we get $φ_{2} = 2 ψ_{1} - 1 ψ_{2}$ .

@@ Line 22: / Line 22: @@
 <math>\psi_2(c \times C + d \times O) = d</math>
+Now we can ask, given <math>\varphi_1</math>, how can we write it in terms of <math>\psi_1, \psi_2</math>?
+Given the mixture <math>a \times CO_2 + b \times CO</math>, we can write this as <math>(a + b) \times C + (2a + b) \times O</math>. Thus, <math>\psi_1(a \times CO_2 + b \times CO) = a+b</math> and <math>\psi_2(a \times CO_2 + b \times CO) = 2a+b</math>. In other words, our task is to express <math>a</math> in terms of <math>a+b</math> and <math>2a+b</math>. We get <math>(2a+b) - (a+b) = a</math>, so <math>\varphi_1 = -\psi_1 + \psi_2</math>.
+Similarly, we get <math>\varphi_2 = 2\psi_1 -1\psi_2</math>.