Linear Algebra Coordinate Systems Isomorphism This is an excerpt from the book.
Let $B$ be the standard basis of the space $P_3$ of polynomials; that is let $B=\{1,t,t^2,t^3\}$.  A typical element $p$ of $P_3$ has the form
$p(t) = a_0 + a_1t+ a_2t^2 + a_3t^3.$
 Since $p$ is already displayed as a linear combination of the standard basis vectors, we conclude that $[P]B = [a_0,a_1,a_2,a_3]$.
This is the part I don't get. My understanding is if I were to change the basis of something, then I would have a matrix, for instance $B$ represented as $\{b_1,b_2\}$. And using the matrix, I would change $x$ to $[x]B$ by performing $B^{-1} x = [x]B$. None of these steps were taken for the above. Could anybody explain how the polynomial equation suddenly turned into a vector with respect to $B$?

The polynomial is not turned into a vector. The vector $[P]B=[a_0,a_1,a_2,a_3]$ shows the coordinate of $f(x)=a_0+a_1x+a_2x^2+a_3x^3$ when you choose $B=[1,x,x^2,x^3]$ as your basis. As I understand, there is no intention to change the basis.
