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?