Im stuck on how to find a transition matrix for the change of basis between two polynomial bases.
lets say that the bases are A={3, x} and B={x, 4+x}.
I understand a method for finding the transition matrix when the elements of the bases are themselves actual vectors. I can just set up an augmented matrix and use row reduction of the new basis vectors to get the result.
I'm assuming in this case I have to think in more algebraic terms, and Ive read the textbook about 15 times now but its just not making any sense. If someone can explain how to solve this Id be very grateful.
Best Answer
You need to write the elements of one basis in the other one. For example, $$ 3=-\frac34\,x+\frac34\,(4+x),\ \ \ x=1\,x + 0\, (4-x). $$ The transition matrix is then $$ \begin{bmatrix} -\frac34&1\\\frac34&0\end{bmatrix} $$