coordinate

• Nov 12th 2010, 05:08 AM
alexandrabel90
coordinate
im very lost with transition matrix..thought i understand it alr but turns out i haven..

if i define T to be the transition matrix from S to S'
then let (v)_s be the coordinate vector wrt to S...

then its right of me to say that the coord vector wrt to S' is T(v)_s right?

let S= {v_1,...,v_n}, S' = {v'_1,...,v'_n} be the basis for the vector space V
then in my notes, it states that (v_1,...,v_n) = (v'_1,...,v'_n)T. but i thought it should be
T(v_1,...,v_n) = (v'_1,...,v'_n) since T is from S to S'?

i dont get what it means when my prof puts the T behind the vectors.
• Nov 12th 2010, 07:42 AM
FernandoRevilla
Perhaps this example for a vector space $\displaystyle E$ with $\displaystyle \dim E=2$ may help you:

Let $\displaystyle B=\left\{{u_1,u_2}\right\},\;B'=\left\{{u'_1,u'_2} \right\}$ be two basis of $\displaystyle E$.

Suppose:

$\displaystyle \left\{ \begin{array}{c} u'_1=3u_1-2u_2 \\ u'_2=4u_1+5u_2 \end{array}\right$

Denote:

$\displaystyle {}^t(x_1,x_2)$ : coordinates of $\displaystyle x\in E$ in respect to $\displaystyle B$

$\displaystyle {}^t(x'_1,x'_2)$ coordinates of $\displaystyle x\in E$ in respect to $\displaystyle B'$

Then:

$\displaystyle x=x'_1u'_1+x'_2u'_2=x_1u_1+x_2u_2$

Identifying coordinates on $\displaystyle B$, we easily obtain:

$\displaystyle \left({\begin{array}{ccc}{x_1}\\{x_2}\end{array}\r ight)=\begin{pmatrix}{3}&{4}\\{-2}&{5}\end{pmatrix}\left({\begin{array}{ccc}{x'_1} \\{x'_2}\end{array}\right)$

Or equivalently:

$\displaystyle \left({x_1},\;{x_2}\right)=\left({x'_1},\;{x'_2}\r ight)\begin{pmatrix}{3}&{-2}\\{4}&{5}\end{pmatrix}=(x_1',x_2')T$

It's easy to generalize for $\displaystyle \dim E=n$ .

Regards.