# Transition Matrix of T

1. Jun 25, 2017

### SetepenSeth

1. The problem statement, all variables and given/known data
Find the transition matrix $P$ of a transformation defined as
$T:ℝ_2→ℝ_3$
$T:\begin{bmatrix}a\\b\end{bmatrix} = \begin{bmatrix}a+2b\\-a\\b\end{bmatrix}$

For basis

$B=\begin{bmatrix}1\\2\end{bmatrix},\begin{bmatrix}3\\-1\end{bmatrix}$

$C=\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{bmatrix}1\\1\\1\end{bmatrix}$

Verify this matrix by calculating $Pv$ and comparing the result with the actual transformation $T(v)$ for

$v=\begin{bmatrix}-7\\7\end{bmatrix}_B$

2. Relevant equations

$[x]_C=P_C←_B[x]_B$

3. The attempt at a solution

When applying $T$ the results shows

$T(v)=\begin{bmatrix}7\\7\\7\end{bmatrix}$

However, my attempts to find the transition matrix $P$ have been unsuccessful, apparently my struggle is with the fact that $ℝ_2$ maps to $ℝ_3$, therefore the transition matrix $P$ has to be $3x2$, thus non invertible, so I can't find it through its inverse, I attempted separating the transformation into its components like

$T(v)= a \begin{bmatrix}1\\-1\\0\end{bmatrix} + b\begin{bmatrix}2\\0\\1\end{bmatrix}$

Or $T(B_1), T(B_2)$

$T(B_1)= \begin{bmatrix}5\\-1\\2\end{bmatrix}$
$T(B_2)= \begin{bmatrix}1\\-3\\-1\end{bmatrix}$

But neither $\begin{bmatrix}1&2\\-1&0\\0&1\end{bmatrix}$ nor $\begin{bmatrix}5&1\\-1&-3\\2&-1\end{bmatrix}$ are the same as on my answer key therefore I believe my approach to this matrix is flawed.

2. Jun 26, 2017

### WWGD

Usually transition matrix is between different spaces for the same vector space. How are you defining it?

3. Jun 26, 2017

### haruspex

I agree with WWGD that normally "transition matrix" means a mapping from and to the same vector space. Putting that aside...

I did not follow your method. I would write B as (b1, b2) etc. then see how to write $\begin{bmatrix}a\\b\end{bmatrix}$ as a linear combination of the vectors b1, b2. Similarly, how to write $\begin{bmatrix}a+2b\\-a\\b\end{bmatrix}$ in terms of the ci basis vectors.

4. Jun 27, 2017

### SetepenSeth

Thank you both.

Indeed my problem is that I was missing the step to write the vectors in terms of Ci basis.

However I've found I have a conceptual mistake in my question. Apparently, a transition matrix is completely different from a matrix associated to a transformation, and it was the later the one I was looking for. How different are these two?

Last edited: Jun 27, 2017
5. Jun 27, 2017

### haruspex

Transition suggests a change of state within a system. In a vector space context that would mean a transition from one state vector to another state vector within the same space.