What is the geometric meaning of the transformation of R2/R3 when every vector is multiplied by −1? Is it a rotation? I'd imagine a sphere with the center at the origin and all length of the vectors equals the radius. But I can't imagine what would happen if all the vectors is multiplied by -1, what would it be?
I would assume its a rotation, but my teacher says its not. Why?
In the case of R2, would it be a rotation?
 A: Multiplication by $-1$ is a rotation in $\mathbf R^n$ if and only if $n$ is even. If $n$ is odd, the determinant is $-1$, so the transformation is orientation-reversing and therefore cannot be a rotation.
A: I would say that conceptually this is not a rotation: however it turns out that it is essentially the same as a rotation in $\Bbb R^2$ - but not in $\Bbb R^3$.
To visualise what multiplying the vector by $-1$ means I would suggest the following: take your vector $\bf v$ and multiply it by a scalar which starts at $1$ and gradually decreases.  This means the vector will retain the same direction but will gradually get shorter.  As the scalar becomes zero the vector shrinks to nothing.  But if the scalar continues to decrease - so now it is negative - the vector starts to increase in length again, but in the opposite of the original direction.  When the scalar gets to $-1$ you have a vector with the same length as the original, but in the opposite direction.  So, think of the transformation as moving the tip of the vector "through the origin to the other side".
Now this is very easy to draw in $\Bbb R^2$ (please try it!) and you can see that it is the same effect as simply rotating the given vector through $180^\circ$.  It's harder to draw in $\Bbb R^3$ and you might think it's the same, but. . . 
Visualise the axes of $\Bbb R^3$ by using the thumb and first two fingers of your right hand, with the middle finger bent at right angles.  If you can imagine "reversing all these" through the origin, you will get a left hand, not a right hand.  There is no way you can rotate your right hand so that it becomes a left hand - or if there is, you need urgent medical assistance ;-) - and this transformation is a reflection, not a rotation.
Hope this helps!
