The definition you told is fine. However, the notion you have about the components of a vector is not ok completely. Actually the components are generally said and the representation you are having, talks about rectangular components of a vector. As the coordinate system changes the rectangular components of a vector also chnage their direction accordingly keeping the magnitude fixed.
If you are talking about a vector, the magnitude remains changed irrespective of the coordinate system. However, as you change the coordinate system or may be if you are rotating the coordinate system then the direction changes and accordingly the direction of the rectangular components change.
I hope this helps!!!!!!!!!