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23 lines
812 B
TeX
23 lines
812 B
TeX
Relevant definitions used throughout Analysis II.
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\subtext{$\textbf{A} \in \R^{m \times n},\quad x,y \in \R^n,\quad \alpha \in \R$}
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\definition \textbf{Scalar Product} $x \cdot y :=\sum_{i=0}^{n} (x_i \cdot y_i)$
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\definition \textbf{Euclidian Norm} $||x|| := \displaystyle\sqrt{\sum_{i=1}^{n} x_i^2}$\\
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\subtext{Used to generalize $|x|$ in many Analysis I definitions}
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\lemma \textbf{Properties of} $||x||$
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\begin{center}
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$
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\begin{array}{ll}
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(i) & ||x|| \geq 0 \\
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(ii) & ||x|| \iff x = 0 \\
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(iii) & ||\alpha x|| = \alpha \cdot ||x|| \\
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(iv) & ||x + y|| \leq ||x|| + ||y||\quad \text{(Triangle Inequality)}
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\end{array}
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$
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\end{center}
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\definition \textbf{Trace} $\text{Tr}(\textbf{A}) := \displaystyle\sum_{i=0}^{\text{min}(m,n)} \textbf{A}_{i, i}$
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