**My problem:**
Suppose

1)Matrix for linear transformation

relative to basis

2)Matrix for linear transformation

relative to basis

3)Transition matrix from basis

to basis

is:

4)Transition matrix from basis

to basis

is :

Now we know that:

And

where

is coordinate matrix of

relative to basis

and

is coordinate matrix of

relative to

Now the book(Elementary Linear Algebra by Larson 6th edition) I am reading states that:

"But by the definition of the matrix of a linear transformation relative to a basis, this implies that:

"

**My question:**
My question is: why's that?

I know from above that:

which makes:

But what reasoning makes the statement

true? How did the author deduced this to be true?