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[IML] Transformers

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RobinB27
2026-06-17 17:58:56 +02:00
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semester6/iml/parts/0a_linalg.tex
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\smalltext{We only consider $\R^n$}
\definition \textbf{Linear Subspace} $\mathcal{V} \subseteq \R^n$\\
\smalltext{s.t. $\forall \mathbf{u_1},\mathbf{u_2} \in \mathcal{V},\quad \forall \alpha, \beta \in \R:\quad \alpha \mathbf{u_1} + \beta \mathbf{u_2} \in \mathcal{V}$}
\definition \textbf{Linear Independence} $\{v_1,,\ldots,v_k\} \in\R^n$\\
\smalltext{s.t. $\sum_{i=1}^{k}c_iv_i = 0 \iff c_1 = \cdots = c_k = 0$}
\definition \textbf{Dimension} $\dim(\mathcal{V}) \in \N$\\
\smalltext{Max. number of linearly independent vectors in $\mathcal{V}$}
\definition \textbf{Span} $\text{span}(v_1,\ldots,v_k) := \Bigl\{ \sum_{i=1}^{k}c_iv_i \sep c_i \in \R \Bigr\}$\\
\subtext{This is a linear subspace of $\{v_1,\ldots,v_k\}$}