Prove that 3d rotation is linear

Question

In a 2d space, a transformation is linear if $f(v+w) = f(v) + f(w)$ and $f(kv) = k*f(v)$, and rotation preserves addition so it is linear. In a 3d space, similar rules apply: $(x, y, z) + (l, j, k) = (x + t, y + j, z + k)$, and $k(x, y, z) = (kx, ky, kz)$.

It follows that applying a given rotation matrix in 3d space

(for example $M = \begin{bmatrix} 1 & 0 & 0 \\ 0 & cos\theta & -sin\theta \\ 0 & sin\theta & cos\theta\end{bmatrix} $ )

should also be linear. The above is just a rotation along the x-axis so it seems trivial. How can I prove that all 3d rotations are linear, given a rotation matrix M?

Edit

How can I prove that all 3d rotations are linear using the definition of linearity, ie using multiplication and addition? Can I use the fact that rotation itself is multiplication, and it preserves addition?

Answer 1

Any rotation is a rotation around some axis. You can write it as a change of basis matrix times a standard rotation matrix (similar to $R_x$, but just being around another non standard axis) in the new basis, then back to the standard basis. So it is just matrix multiplication, and matrix multiplication is linear.

$M$ is a rotation matrix and generates a linear transformation $T$. It operates on vectors $v$ by $T(v)=Mv$. Matrix multiplication is linear, so $T$ is a linear transformation.

Answer 2

I think this question has nothing to do with bases and matrix representations. This question is an excellent example of treating linear transformations on their own terms rather than analyzing their component space representations.

A transformation is linear if it doesn't matter if you add two vectors first and then transform the sum or if you transform each vector first and then add together the results + and the same for multiplication by a scalar. Here's a video that explains this: https://www.lem.ma/17

And here's a video that shows that rotations are linear: https://www.lem.ma/gs The video is for rotations in the plane, but there is no difference as far as linearity is concerned.