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I've never really needed to use linear algebra, but I did take a class on it recently, so... Why does it matter that it's unit vectors in a new coordinate syst

I've never really needed to use linear algebra, but I did take a class on it recently, so... Why does it matter that it's unit vectors in a new coordinate system? I don't really care about the new coordinate system, I care about rotating something to this position in my old coordinate system. Saying "oh yeah it's <1,1,1> in this coordinate system" doesn't really help because I still need to translate <1,1,1> in the old coordinate system to <1,1,1> in the new coordinate system... and now I've got the same problem again. I mean, I get that a rotation matrix, or any matrix for that matter, represents unit vectors in a new coordinate system. If I got nothing else from that course I at least learned that a matrix represents like 50 different concepts like systems of equations, coordinate spaces, etc. A big part of my course was applying rules for one representation of a problem to another equivalent representation to a problem. What I never understood is why it matters how you're representing the problem. Identifying that a rotation matrix is just unit vectors in a new coordinate space doesn't help me solve my problem of wanting to rotate something, yet it felt like 75% of my course was about converting problems from one representation to another.

Twisol 6y ago
I think the idea is to recontextualize the coordinate system of your object as not being the global coordinate system in the first place. The matrix defines the orientation of your object relative to the global coordinate system. The question of "how to rotate something" rests on operational semantics. The question of "how is this related to that" rests on denotational semantics, and is much more general. The beauty of linear algebra is how closely the two are related: every linear transformation (the relationship) gives a matrix given a choice of basis, and matrices have a direct algorithm for transforming coordinate vectors.
overgard 6y ago
In this case, the idea is to make it easier to debug by looking at it. If you work in 3D graphics, usually looking at euler angles (ie, rotation around x,y,z) is pretty intuitive, but it has a lot of issues with gimbal lock, order the rotations happen in, etc. Quaternion's fix that, but they're very unintuitive to look at (well, unless there's another tricky like this I'm unaware off -- I'm sure there is). On the other hand, this trick with the rotation matrix lets me represent an arbitrary rotation, but it's also pretty easy to take that representation back to something I can understand. In 3d graphics (or physics or what have you) you're constantly moving between coordinate systems/frames of reference because certain things are very easy to represent in a local coordinate system but are difficult to work with in a global one
fifticon 6y ago
But that is actually very useful problem solving. Identifying a problem in your hand as identical to a different problem you know solutions for, allows you to solve it effortlessly like one of those impossible smith puzzles with knots and strings.

Twisol 6y ago

I think the idea is to recontextualize the coordinate system of your object as not being the global coordinate system in the first place. The matrix defines the orientation of your object relative to the global coordinate system. The question of "how to rotate something" rests on operational semantics. The question of "how is this related to that" rests on denotational semantics, and is much more general. The beauty of linear algebra is how closely the two are related: every linear transformation (the relationship) gives a matrix given a choice of basis, and matrices have a direct algorithm for transforming coordinate vectors.

overgard 6y ago

In this case, the idea is to make it easier to debug by looking at it. If you work in 3D graphics, usually looking at euler angles (ie, rotation around x,y,z) is pretty intuitive, but it has a lot of issues with gimbal lock, order the rotations happen in, etc. Quaternion's fix that, but they're very unintuitive to look at (well, unless there's another tricky like this I'm unaware off -- I'm sure there is). On the other hand, this trick with the rotation matrix lets me represent an arbitrary rotation, but it's also pretty easy to take that representation back to something I can understand. In 3d graphics (or physics or what have you) you're constantly moving between coordinate systems/frames of reference because certain things are very easy to represent in a local coordinate system but are difficult to work with in a global one

fifticon 6y ago

But that is actually very useful problem solving. Identifying a problem in your hand as identical to a different problem you know solutions for, allows you to solve it effortlessly like one of those impossible smith puzzles with knots and strings.