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How do you prove that rotation of a vector is a linear transformation?

It's intuitive (although not completely crystal clear to me) that it is a linear transformation at the 2d level, but how do I prove it to myself (that this is a general property of rotations)?

For example, rotate vector ##\vec{V}## in the xy plane by 30º. ##Rot(\vec{V})=Rot(Vx \vec{e1} + Vy \vec{e2} + Vz \vec{e3})= Vx Rot(\vec{e1}) +Vy Rot(\vec{e2}) + Vz Rot(\vec{e3}) ##

Or in other words, if I have to rotate a vector on the plane xy (for example), how do I prove that this rotation can be done by rotating only my base vectors (or my axis, if you will) and then drawing my original vector with that "new" base or coordinate axis.

Thanks.

It's intuitive (although not completely crystal clear to me) that it is a linear transformation at the 2d level, but how do I prove it to myself (that this is a general property of rotations)?

For example, rotate vector ##\vec{V}## in the xy plane by 30º. ##Rot(\vec{V})=Rot(Vx \vec{e1} + Vy \vec{e2} + Vz \vec{e3})= Vx Rot(\vec{e1}) +Vy Rot(\vec{e2}) + Vz Rot(\vec{e3}) ##

Or in other words, if I have to rotate a vector on the plane xy (for example), how do I prove that this rotation can be done by rotating only my base vectors (or my axis, if you will) and then drawing my original vector with that "new" base or coordinate axis.

Thanks.

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