Gaining Intuition: Linear Transforms & Coordinate Functions

[Septimra]

So I been working with quaternions as you all know. I get them basically, but to really understand their rotation properties i decided to better understand matrices and how they relate to real valued functions.

a matrix is a transformation of a vector from one vector space to another through a linear transformation.

lets say we are going from ℝ³ to ℝ²

f(v) = f(vx+vy+vz) = f(vx) + f(vy) + f(vz)

This is a linear transformation so these properties can be exploited.

Now we must define how the basis vectors of a vector in a certain vector space are transformed.

The basis vectors can be written as a linear combination of the basis vectors of the vector space as well.

vx = 1vx + 0vy + 0vz

vy = 0vx + 1vy + 0vz

vz = 0vx + 0vy + 1vz

The linear transformation transforms these basis vectors into a linear combination of these basis vectors.

f(vx) = 2w1 + 5w2

f(vy) = 3w1 + 7w2

f(vz) = 4w1

Finally my question is how would you write that if as a coordinate function f(vx, vy, vz) = ?

That equation would then have to satisify all the basis vectors as well

f(1,0,0) = ?

f(0,1,0) = ?

f(0,0,1) = ?

so when you add all these values together you should get the same as a matrix multiplication of

[2 3 4] *[1]
[5 7 0] [1]
[1]

I have no idea how to write it because i have rarely worked with coordinate functions and cannot find much on it online. But i have come to realize that the key to understanding matrices is to understand coordinate functions. Thanks a lot

 

[Simon Bridge]

Knowing how the basis vectors are transformed can help work out what happens with the transformation in general since, then, when you apply the function to a vector, you only need to resolve the vector against the basis.


Since you started with quaternions, you should probably study the 3D rotation matrix.
The main advantage of the quaternion approach is that it requires fewer computational steps - so you use it to build fast-paced video-games. Everywhere else uses the matrix approach.

You'll get a better appreciation of how the different part relate to each oter if you see a concrete example.
Look at:
Tan S.M. (2003) Classical Mechanics http://home.comcast.net/~szemengtan/ClassicalMechanics/SingleParticle.pdf
Section 1.8: "dynamics of a uniformly rotating frame."