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 relative to basis is

2. Matrix for relative to basis is

3. Transition matrix from to is

4. Transition matrix from to is

...there are two ways to get from the coordinate matrix to

the coordinate matrix One way is direct, using the matrix to obtain

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

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

Couldn't it be the case where but ?

Is it possible to kindly tell me the reason how 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?