5 ms·

Funny. Only some months ago I needed to figure out how to represent unit vectors of (a rotated) coordinate system in another coordinate system. After some days

Funny. Only some months ago I needed to figure out how to represent unit vectors of (a rotated) coordinate system in another coordinate system. After some days of drawing on the paper I came to the conclusion that actually the rotation vector columns are just that. But found no way to verify that claim in the internet. Thank you for this.

leereeves 6y ago
That's not only true for rotation matrices. The columns of any matrix are where the unit vectors in the original coordinate system are mapped to when multiplied by the matrix.
- MiroF 6y ago
  What are colleges teaching these days if not this??
  - beefield 6y ago
    I don't know, my college days were a couple of decades ago, so I have already forgotten plenty. But if you can point to a good source in the internet that explains this, I could try to see why my research efforts failed.
    - MiroF 6y ago
      Here's wikipedia. Granted, the mathematicians behind wikipedia seem to be focused on making everything as notationally obscure and hard to understand as possible, but it outlines the concept https://en.wikipedia.org/wiki/Linear_map#Matrices https://en.wikipedia.org/wiki/Linear_map#Matrices
      - beefield 6y ago
        Well, I have to say that if that is the most approachable and easily googled content in the internet to figure this out, I am not too surprised I had to figure it out on my own...
  - exabyte 6y ago
    For me I think the difficulty was just reconciling reality with the concept of a coordinate system. I think it was just some abstract concept that I considered to be out of my cognitive reach of understanding. I just had no way to anchor anything being said to me to a concept I had in my head at the time. I remember this poor teaching fellow where I went to Lehigh University where I studied Mechanical Engineering. He was foreign and had the honor of teaching us Linear Algebra. He must have noticed that none of us had even the slightest damn clue what he was talking about. So, the class before every exam, he would literally just give go over the exam problems nearly verbatim and solve them. I came out of that class with a good grade and not the slightest fucking understanding of Linear Algebra beyond the very superficial algebraic laws of multiplying tensor objects (i.e., scalars (0-D tensor), vectors (1-D tensor), and matrices). I'd like to thank Grant Sanderson (3Blue1Brown) and Mike Cohen (Udemy Linear Algebra course using Matlab and Python) for teaching me linear algebra. And don't forget Gilbert Strang, of course!! (find his course on MIT OCW) The visualizations of the tensors transforming provided by the first two are what really made Linear Algebra start to click for me. I want to go back to Gilbert Strang's course now that I have a better geometric understanding of what is happening and appreciate all his wisdom on the subject.
    - MiroF 6y ago
      As a physics major at the time, I think that learning physics in addition to multivar and linear algebra is so handy in terms of providing reference material for intuitions. Also, learning it as part of a numerical computing stack (like python) where you can experiment and visualize is huge for building intuition.
  - kd0amg 6y ago
    How to compute a matrix product, how to compute the solution to a system of equations, how to compute a determinant (inefficiently), how to compute eigenvalues Lots of how, not enough why
- im3w1l 6y ago
  A really neat thing I learned in physics was Bra-Ket (get it bracket?) notation. Ket's are vectors and written like |u>. Bra's are objects that compute dot products with vectors. They are written like <u|. So <u| |v> = u · v. However if you have |u> <v| then nothing happens. This is a new object. It will first take dot product with v and then multiply with |u> vector. Letting e1, e2.. denote the coordinate basis vectors, we can write a matrix A with coefficients a_ij as the sum Σa_ij |ej> <ei|. This can be rearranged as Σ_i |Σ_j a_ij ej> <ei| Written like this, it is clear that the column is where the unit vector ends up.
dosshell 6y ago
I'm glad you appreciated the text.

leereeves 6y ago

That's not only true for rotation matrices. The columns of any matrix are where the unit vectors in the original coordinate system are mapped to when multiplied by the matrix.

MiroF 6y ago
What are colleges teaching these days if not this??
- beefield 6y ago
  I don't know, my college days were a couple of decades ago, so I have already forgotten plenty. But if you can point to a good source in the internet that explains this, I could try to see why my research efforts failed.
  - MiroF 6y ago
    Here's wikipedia. Granted, the mathematicians behind wikipedia seem to be focused on making everything as notationally obscure and hard to understand as possible, but it outlines the concept https://en.wikipedia.org/wiki/Linear_map#Matrices https://en.wikipedia.org/wiki/Linear_map#Matrices
    - beefield 6y ago
      Well, I have to say that if that is the most approachable and easily googled content in the internet to figure this out, I am not too surprised I had to figure it out on my own...
- exabyte 6y ago
  For me I think the difficulty was just reconciling reality with the concept of a coordinate system. I think it was just some abstract concept that I considered to be out of my cognitive reach of understanding. I just had no way to anchor anything being said to me to a concept I had in my head at the time. I remember this poor teaching fellow where I went to Lehigh University where I studied Mechanical Engineering. He was foreign and had the honor of teaching us Linear Algebra. He must have noticed that none of us had even the slightest damn clue what he was talking about. So, the class before every exam, he would literally just give go over the exam problems nearly verbatim and solve them. I came out of that class with a good grade and not the slightest fucking understanding of Linear Algebra beyond the very superficial algebraic laws of multiplying tensor objects (i.e., scalars (0-D tensor), vectors (1-D tensor), and matrices). I'd like to thank Grant Sanderson (3Blue1Brown) and Mike Cohen (Udemy Linear Algebra course using Matlab and Python) for teaching me linear algebra. And don't forget Gilbert Strang, of course!! (find his course on MIT OCW) The visualizations of the tensors transforming provided by the first two are what really made Linear Algebra start to click for me. I want to go back to Gilbert Strang's course now that I have a better geometric understanding of what is happening and appreciate all his wisdom on the subject.
  - MiroF 6y ago
    As a physics major at the time, I think that learning physics in addition to multivar and linear algebra is so handy in terms of providing reference material for intuitions. Also, learning it as part of a numerical computing stack (like python) where you can experiment and visualize is huge for building intuition.
- kd0amg 6y ago
  How to compute a matrix product, how to compute the solution to a system of equations, how to compute a determinant (inefficiently), how to compute eigenvalues Lots of how, not enough why
im3w1l 6y ago
A really neat thing I learned in physics was Bra-Ket (get it bracket?) notation. Ket's are vectors and written like |u>. Bra's are objects that compute dot products with vectors. They are written like <u|. So <u| |v> = u · v. However if you have |u> <v| then nothing happens. This is a new object. It will first take dot product with v and then multiply with |u> vector. Letting e1, e2.. denote the coordinate basis vectors, we can write a matrix A with coefficients a_ij as the sum Σa_ij |ej> <ei|. This can be rearranged as Σ_i |Σ_j a_ij ej> <ei| Written like this, it is clear that the column is where the unit vector ends up.

dosshell 6y ago

I'm glad you appreciated the text.