I am given a linear transformation $T:\mathbb{R^3} \rightarrow \mathbb{R^3}$
The transformation is linear and is defined by taking a vector in $\mathbb{R^3}$ and rotating it around the axis $L=span{(1,1,0)}$, which is a line in the $(x,y)$ plane, by $\frac{\pi}{4}$
My question: what is the difference between rotating a vector around the x or y or any axis in the $(x,y)$ plane, aren't we just changing the angle it makes with the rotation axis by $\frac{\pi}{4}$?
Edit: To phrase better, what would be the result of a rotation by $\frac{\pi}{4}$of the vector $(1,0,0)$ around the x axis, around the y axis and around the axis $L=span(1,1,0)$.
Rotating "about an axis" means rotating in the plane perpendicular to that axis; the angle of a point with the axis does not change.
For example, the rotation around the $x$-axis by $\pi/4$ is $$ \pmatrix{ 1&0&0\\ 0&1/\sqrt2 & -1/\sqrt2\\ 0&1/\sqrt2& 1/\sqrt2 } $$