Why in some linear transformations, space is flipped?

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Watching 3b1b video on geometric meaning of Matrix transformations, there was a matrix

$\begin{bmatrix}1&3\\2&1\end{bmatrix}$

Which meant that, the $i$ will move to new coordinates $[1 , 2 ]$, which indeed made the x-axis oblique and moved $i$ to $[1,2}]$.

But after that when moving $j$ to $[3,1]$, the space was flipped first and then the $y$ axis were made to rotate so as to move $j$ to $[3,1]$.

I was confused in that. Because if at all we are just to move $i$ and $j$ to new co-ordinates, we can indeed do that without flipping.

What am I missing? please help.

the exact timestamp is 9:20

Thank you !