- #1
Septimra
- 27
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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 alot
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 alot