What does it mean to be the same linear transformation in different bases? I know vectors can be written in different bases, one vector can be equivalent to another in another vector space in different bases, but what is the ideia of equivelent linear transformation in different bases in simple terms ?

We can think to the basis as the reference system to describe the vectors and the idea for the linear transformation $T:V\to W$ is that for


*

*$u_1$ and $v_1$ are vectors in base $B_{V,1}$

*and $w_1=T(au_1+bv_1)=aT(u_1)+bT(v_1)$ vector in base $B_{W,1}$


we have that for


*

*$u_2$ and $v_2$ vectors in base $B_{V,2}$ 

*and $w_2=T(au_2+bv_2)=aT(u_2)+bT(v_2)$ vector in base $B_{W,2}$


we have that


*

*if $u_2$ and $v_2$ correspond to $u_1$ and $v_1$ in base $B_{V,2}$ then $w_2$ corresponds to $w_1$ in base $B_{W,2}$

