I have come across two questions about matrices and changes of bases. They seem to be the same question, but require different approaches. I can't figure out why.

First question can be found at: Given a Transformation Matrix $T$, find $T$ relative to a new basis $\beta$

It says $T(a_1,a_2,a_3) = (3a_1+a_2,a_1+a_3,a_1-a_3)$. $(a_1,a_2,a_3)^T$ is written with regards to the standard basis.

$\beta$ is a new basis = $\{(1,0,0), (1,1,0), (1,1,1)\}$.

I want to write $T$ w.r.t $\beta$. To do this, I do $TC$, where $C = \left(\begin{array}{ccc}1 & 1 & 1 \\0& 1 & 1 \\0 & 0 & 1\end{array}\right)$. I get the right answer.

My second question is let $T'(x,y) = (4x -2y, 2x+y)$. Let $\beta'=\{ (1,1),(-1,0)\}$. I want to write $T'$ in the given basis, $\beta'$.

My approach was to do the same thing as above, $T'C'$, where $C' = \left(\begin{array}{cc}1 & -1 \\1 & 0\end{array}\right)$. I do NOT get the right answer. $C'^{-1}TC'$ does give me the right answer.

Why does one approach give me the right answer one time but the wrong answer the next time? How do I decide which approach to use?


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