About Linear Transformation (Linear transform and its matrix representation) Suppose I found that in some basis, call it $B$,
$[T]_B = [T^*]_B$, where $[T]_B$ and $[T^*]_B$ are the matrix representation of $T$ and $T^*$ in the basis $B$.
Can I say that then $T = T^*$?
 A: If I understood your question correctly, you're wondering whether if we have two linear maps $$T,T^*\in \mathcal{L}(V,W)$$ (the vector space of linear maps from $V$ to $W$) and with respect to some basis $\mathcal{B}$ they have the same matrix representation, are they the same linear map with respect to any basis?
The answer in short is yes. 
This is because the matrix of a linear map fully characterizes the action of the linear map acting on a given vector space. We may change matrix representations by applying change of basis operators to the bases of $V$ or $W$, but the action would be symmetric on both the matrix representations of $T, T^*$.
A: Yes, because two functions are equal precisely when they have the same domain and are equal at every input.
The value of a linear transformation at any input is uniquely determined by its value on a set of vectors that make up a basis, and to say that the two matrices are equal means that the transformations agree on basis vectors comprising $B$.
