41 lines
1.7 KiB
TeX
41 lines
1.7 KiB
TeX
\section{The Change of Coordinate Matrix}
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\begin{theorem}\label{Theorem 2.22}
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\hfill\\
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Let $\beta$ and $\beta'$ be two ordered bases for a finite-dimensional vector pace $V$, and let $Q = [I_V]_{\beta'}^\beta$. Then
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\begin{enumerate}
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\item $Q$ is invertible.
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\item For any $v \in V$, $[v]_\beta = Q[v]_{\beta'}$.
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\end{enumerate}
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\end{theorem}
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\begin{definition}
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\hfill\\
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The matrix $Q=[I_V]_{\beta'}^\beta$, defined in \autoref{Theorem 2.22}, is called a \textbf{change of coordinate matrix}. Because of part (2) of the theorem, we say that \textbf{$Q$ changes $\beta'$-coordinates into $\beta$-coordinates}.
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\end{definition}
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\begin{definition}
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\hfill\\
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A linear transformation that maps a vector space $V$ into itself is called a \textbf{linear operator on $V$}.
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\end{definition}
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\begin{theorem}
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\hfill\\
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Let $T$ be a linear operator on a finite-dimensional vector space $V$, and let $\beta$ and $\beta'$ be ordered bases for $V$. Suppose that $Q$ is the change of coordinate matrix that changes $\beta'$-coordinates into $\beta$-coordinates. Then
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\[[T]_{\beta'}=Q^{-1}[T]_\beta Q\]
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\end{theorem}
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\begin{corollary}\label{Corollary 2.8}
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\hfill\\
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Let $A \in M_{n \times n}(\F)$, and let $\gamma$ be an ordered basis for $\F^n$. Then $[L_A]_\gamma = Q^{-1}AQ$, where $Q$ is the $n \times n$ matrix whose $j$th column is the $j$th vector of $\gamma$.
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\end{corollary}
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\begin{definition}
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\hfill\\
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Let $A$ and $B$ be matrices in $M_{n \times n}(\F)$. We say that $B$ is \textbf{similar} to $A$ if there exists an invertible matrix $Q$ such that $B = Q^{-1}AQ$.\\
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Notice that the relation of similarity is an equivalence relation. So we need only say that $A$ and $B$ are similar.
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\end{definition}
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