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Definition
Suppose \(T:V \to W\) is a linear transformation and \(B,B'\) are bases for \(V,W\), resp. Let \(B = \{v_1,\ldots,v_{n} \}\) and \(B' = \{w_1,\ldots,w_{m} \}\).
The *matrix of T with respect to the bases \(B,B'\) is the \(m \times n\) matrix \([T]\(B\)’\) whose \(j\)th column is \([Tv_{j}]_{B'}\). In other words, if
\begin{align*} Tv_{j} = \sum_{i=1}^{m} a_{ij}w_{i}, \end{align*}then \([T]_{B,B'} = (a_{ij})\).
\([Tv_{j}]_{B'}\) is the coordinate vector of \(Tv_{j}\) in the basis \(B'\).
\begin{align*} [Tv_{j}]_{B'} = [a_{1j} a_{2j} \cdots a_{mj}]^T. \end{align*}Theorems
Exer 2.2 If \(V\) is an \(n\)-dimensional vector space and \(B\) a
basis. Then the map \(F:\text{End}(V) \to M_{n}(\mathbb{C})\) \(F(T) =
[T]_{B}\) is an isomorphism of unital rings.
\[
([T^*]_{B})_{ij} = (i^{th} \text{ entry of } [T^*v_{j}]_{B}) = \langle T^* v_{j}, v_{i} \rangle.
\]
If \(B\) is an orthonormal basis of an inner product space \(V\), and if
\(T \in \text{End}(V)\), then
\[
[T^*]_{B} = ([T]_{B})^*.
\]
Suppose \(V\) is an \(n\)-dimensional vector space and \(T \in
\text{End}(V)\). Then \([T]_{B}\) self-adjoint \(\implies\) \(T\) is is self
adjoint.
Let \(B\) be the standard basis of \(V = \mathbb{C}^n\).
Let \(T \in \text{End}(\mathbb{C})\) be the linear map given by left
multiplication by \(A \in M_{n}(\mathbb{C})\). Then \([T]_{B} = A\).