# Math Help - coordinate

1. ## 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.

2. Perhaps this example for a vector space $E$ with $\dim E=2$ may help you:

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

Suppose:

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

Denote:

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

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

Then:

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

Identifying coordinates on $B$, we easily obtain:

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

Or equivalently:

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

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

Regards.