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===Active transformation===
As an active transformation, <math>T</math> transforms the initial vector <math>\mathbf{v}=(v_x,v_y,v_z)</math> into a new vector <math>\mathbf{v}'=(v'_x,v'_y,v'_z)=T\mathbf{v}=T(v_x,
If one views <math>\{\mathbf{e}'_x=T(1,0,0),\ \mathbf{e}'_y=T(0,1,0),\ \mathbf{e}'_z=T(0,0,1)\}</math> as a new basis, then the coordinates of the new vector <math>\mathbf{v}'=v_x\mathbf{e}'_x+v_y\mathbf{e}'_y+v_z\mathbf{e}'_z</math> in the new basis are the same as those of <math>\mathbf{v}=v_x\mathbf{e}_x+v_y\mathbf{e}_y+v_z\mathbf{e}_z</math> in the original basis. Note that active transformations make sense even as a linear transformation into a different vector space. It makes sense to write the new vector in the unprimed basis (as above) only when the transformation is from the space into itself.
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