First I'll give an excerpt from Larson's Linear Algebra book and then I'll ask my question. It is something like this:

1. Matrix for $\displaystyle T$ relative to basis $\displaystyle B$ is $\displaystyle A$

2. Matrix for $\displaystyle T$ relative to basis $\displaystyle B'$ is $\displaystyle A'$

3. Transition matrix from $\displaystyle B'$ to $\displaystyle B$ is $\displaystyle P$

4. Transition matrix from $\displaystyle B$ to $\displaystyle B'$ is $\displaystyle P^{-1}$

...there are two ways to get from the coordinate matrix $\displaystyle [v]_{B'}$ to

the coordinate matrix $\displaystyle [T(v)]_{B'}$ One way is direct, using the matrix $\displaystyle A'$ to obtain

$\displaystyle A'[v]_{B'} = [T(v)]_{B'}$

The other way is indirect, using the matrices and to obtain

$\displaystyle P^{-1}AP[v]_{B'} = [T(v)]_{B'}$

But by the definition of the matrix of a linear transformation relative to a basis this implies that: <---My question about this sentence is given below

$\displaystyle A' = P^{-1}AP$

Couldn't it be the case where $\displaystyle P^{-1}AP[v]_{B'} = [T(v)]_{B'}$ but $\displaystyle P^{-1}AP \neq A'$?

Is it possible to kindly tell me the reason how $\displaystyle A' = P^{-1}AP$ follows from the definition of linear transformation relative to a basis?

Does it mean that if two linear transformations are equal their transformation matrices are also equal?