I was watching this video. (Actually, I watched it 3 times because I couldn't understand it.) And right then, he showed that the trasformation matrix for rotation and then sheer is
I understood how he got the Rotation Matrix, that was the easy part, but how would you describe the Sheer matrix? Like, are the new coordinates dependent on the newly formed cartisian plane after Rotaiton or is it based on the initial cartian system. I tries both of them, but none seems to be the exact answer for final transformation. Infact, I get the transpose of the matrix instead.
Can you explain how he got the Sheer Matrix.
The shear transformation, when acting on its own (without a rotation in front), sends the first basis vector to itself (i.e. $\left[\begin{smallmatrix}1\\0\end{smallmatrix}\right]$), and the second basis vector to $\left[\begin{smallmatrix}1\\1\end{smallmatrix}\right]$. So that's how you get the columns of the matrix that represents the shear transformation.
This follows exactly the same system for determining the columns of the matrix representation of a linear transformation that he established in (if I recall correctly) video 3.
And there is no "old" or "new" coordinates here. It's the same coordinates all the way through (you will see in the video that the coordinate axes stay put). It is the plane that is transformed "below" these coordinates. The shear transformation does the same thing to the plane regardless of whether it has been previously rotated: keeps all horizontal lines horizontal and makes all vertical lines diagonal.