What Is the Transition Matrix for T in This Transformation?

[SetepenSeth]

Homework Statement

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

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

Homework Equations

$[x]_C=P_C←_B[x]_B$

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.

Any advise would be appreciated.

 

[WWGD]

Homework Statement

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

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

Homework Equations

$[x]_C=P_C←_B[x]_B$

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.

Any advise would be appreciated.

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

 

[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 c_i basis vectors.

 

[SetepenSeth]

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 c_i basis vectors.

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?

 

[haruspex]

a transition matrix is completely different from a matrix associated to a transformation

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.