34 lines
938 B
TeX
34 lines
938 B
TeX
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\section{Matrices}
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\theorem{}<thebigtheorem>
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Any linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as an $n \times m$ matrix. \\
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Conversely, every $n \times m$ matrix represents a linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ \\
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\vspace{2mm}
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In other words, \textbf{matrices are linear transformations}. \\
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The next two problems provide a proof.
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\problem{}<prooffwd>
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Let $A$ be an $m \times n$ matrix, and $v$ an $m \times 1$ vector. \\
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Show that the map $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(v) = Av$ is linear. \\
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\vfill
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\problem{}<proofback>
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Show that any linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as $T(v) = Av$.
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\vfill
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\pagebreak
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\problem{}
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Consider the transformation $D: \mathbb{P}^3 \to \mathbb{P}^2$ defined by $D(p) = \frac{d}{dx}(p)$. \\
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Find a matrix that corresponds to $D$. \\
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\hint{$\mathbb{P}^3$ and $\mathbb{R}^4$ are isomorphic. How so?}
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\vfill
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\pagebreak
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