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.(adsbygoogle = window.adsbygoogle || []).push({});

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

lets say we are going from ℝ^{3}to ℝ^{2}

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Gaining Intuition

**Physics Forums | Science Articles, Homework Help, Discussion**