Comments on transpose matrix and general matrix is completely messed up
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transpose matrix and general matrix is completely messed up
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 :
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.
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I suspect what's tripping you up here is the difference between applying a linear transformation to a vector versus applying a change of basis. Both involve matrix multiplication, but the nature of the transformation is different.
First, some words about vectors. There are two ways to introduce students to the concept of vectors. The first is that a vector is just a pair or triple or $n$-tuple of scalars. Operations like vector addition and scalar multiplication are defined in terms of those scalars. The second way is to think of vectors as abstract things that obey axioms involving vector addition and scalar multiplication. For finite-dimensional vector spaces, everything that's true in one presentation is true in the other. However, the first presentation assumes some standard basis, which acts as a way to assign coordinates to vectors. Much like a distance of ‘5’ only makes sense if you know that it's implicitly ‘5 meters’ or ‘5 lightyears’, a vector $\left[\begin{smallmatrix}2\\7\end{smallmatrix}\right]$ only makes sense if you know (or assume that we all agree on) what vector corresponds to ‘1’ of each of those coordinates. That set of vectors is the basis. In the second presentation, a basis is just an arbitrary collection of vectors $\vec{e_i}$ that span the space (meaning that any vector is some linear combination of the basis vectors).
You can't use matrices to represent abstract vectors without implicitly using some basis. When we write that a vector $\vec{v} = \left[\begin{smallmatrix}v_1\\v_2\end{smallmatrix}\right]$, what we are actually saying is that $\vec{v} = v_1\vec{e_1} + v_2\vec{e_2}$, where $\vec{e_i}$ are the elements of our implicit basis. We can abuse notation a little bit and write this as $\vec{v} = \left[\begin{smallmatrix}\vec{e_1} & \vec{e_2}\end{smallmatrix}\right]\left[\begin{smallmatrix}v_1\\v_2\end{smallmatrix}\right]$. (All this can be made rigorous without too much more work, but I'm trying to keep this as simple as I can.)
Now suppose you want to represent $\vec{v}$ in another basis $\vec{b_i}$. $\vec{v}$ remains the same vector, so you want to find $w_i$ such that $\vec{v} = \left[\begin{smallmatrix}\vec{e_1} & \vec{e_2}\end{smallmatrix}\right]\left[\begin{smallmatrix}v_1\\v_2\end{smallmatrix}\right] = \left[\begin{smallmatrix}\vec{b_1} & \vec{b_2}\end{smallmatrix}\right]\left[\begin{smallmatrix}w_1\\w_2\end{smallmatrix}\right]$. If you know the $\vec{b_i}$ in terms of the $\vec{e_i}$, then you can expand them—if $\vec{b_i} = b_{i1}\vec{e_1} + b_{i2}\vec{e_2}$, then: $$ \begin{align} \vec{v} &= \begin{bmatrix}b_{11}\vec{e_1} + b_{12}\vec{e_2} & b_{21}\vec{e_1} + b_{22}\vec{e_2}\end{bmatrix}\begin{bmatrix}w_1\\w_2\end{bmatrix} \\ &= \begin{bmatrix}\vec{e_1} & \vec{e_2}\end{bmatrix}\begin{bmatrix}b_{11} & b_{21}\\b_{12} & b_{22}\end{bmatrix}\begin{bmatrix}w_1\\w_2\end{bmatrix} \end{align} $$
That middle matrix is the change of basis matrix. Notice that its columns are the basis vectors $\vec{b_i}$ when expressed in the standard basis. When you use a matrix to transform a vector, you multiply the vector to the right of the matrix; but notice that when you use this change of basis matrix to transform a basis, the basis appears to the left of the matrix. Same fundamental operation—matrix multiplication—but two different ways to use it depending on what you're doing.
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