I was studying Matrix. I never attend any lecture. So I don't have much more idea of Matrix. I had learned multiplication, addition, subtraction and inverse matrix last year from some YT tutorial. I never learned the conversion of algebraic equation to Matrix or vice-versa.

I had found it in my book last few days ago. And I had watched a similar video on the same topic in YT. I was wondering both had gave different physical (algebraic expression) meaning.

My Book example :

$$\epsilon=\begin{bmatrix}0 & d\Omega_3 & -d\Omega_2 \\ -d\Omega_3 & 0 & d\Omega_1 \\ d\Omega_2 & -d\Omega_1 & 0 \end{bmatrix}$$

$$d x_1 = x_2 d\Omega_3 − x_3 d\Omega_2$$

Of course not too interesting..! Here they just took some values from first row. But the problem arise when I go to YT.

YT example :

Video link at accurate time

He wrote that (I can't write tilde above of vector $e$ hence I am using $d$ here)$$\vec{d_1}=2\vec{e_1}+1\vec{e}$$ $$\vec{d_2}=-\frac{1}{2}\vec{e_1}+\frac{1}{4}\vec{e_2}$$

$$F=\begin{bmatrix}2 & -\frac{1}{2} \\ 1 & \frac{1}{4}\end{bmatrix}$$

According to that author, first line should be in row if that was transpose matrix. But in Goldstein's Classical Mechanics book transpose matrix is opposite of it.

I think there's something that I am missing. Or that video had confused me cause, in his last video he just wrote what I learned later he wrote in title, description and comment that he just made a simple mistake. He just said it is the correct one.