- #1
SetepenSeth
- 16
- 0
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.