I have read on this site and from various sources trying to get an understanding of what the change of basis matrix and change of basis operator is. Below, I have given the definitions as I understand them and hope that someone can either confirm or correct my understanding of these things.
Part I
Let $v := (v_1, \dots, v_n)$ and $w := (w_1, \dots, w_n)$ be two bases of a vector space $V$. By a theorem, there exists a unique isomorphism $T:V \rightarrow V$ given by $T(v_i) = w_i, i=1, \dots n$. Let $[T]^v_w$ denote the matrix of $T$ with respect to $v$ and $w$. Then for any $x \in V$, by another theorem, $ [T]^v_w[x]_v = [x]_w $ where $[x]_b$ denotes the column matrix of $x$ with respect to a basis $b$. The matrix $[T]^v_w$ is called the change of basis matrix or the transition matrix from $v$ to $w$ and $T$ is called the transition automorphism or transition operator from $v$ to $w$.
Question: Is everything in the above acceptably defined?
Part II Looking at the above componentwise, for a vector $x = a^1v_1 + \dots +a^nv_n$ expressed in terms of the basis $v$, the $k$-th coordinate of $[x]_w$ is given by $ ([x]_w)_k = \sum_{j = 1}^n T^k_ja^j $ where $T^k_j$ denotes the entry of $T$ that lives in the $k$-th row and $j$-th column of $[T]^v_w$.
Question: How are the actual basis vectors related to one another with respect to the components of $T$? I think it should be $w_k = \sum_{k=1}^n T^j_k v_j$. How does one prove this rigorously? I believe it follows from the fact that the $T(v_k)$ corresponds to the $k^th$ column of $T$ which is represented by the scalars $T^j_k, k=1, \dots, n$. Is this right?