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15 lines
728 B
TeX
15 lines
728 B
TeX
\smalltext{We only consider $\R^n$}
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\definition \textbf{Linear Subspace} $\mathcal{V} \subseteq \R^n$\\
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\smalltext{s.t. $\forall \mathbf{u_1},\mathbf{u_2} \in \mathcal{V},\quad \forall \alpha, \beta \in \R:\quad \alpha \mathbf{u_1} + \beta \mathbf{u_2} \in \mathcal{V}$}
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\definition \textbf{Linear Independence} $\{v_1,,\ldots,v_k\} \in\R^n$\\
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\smalltext{s.t. $\sum_{i=1}^{k}c_iv_i = 0 \iff c_1 = \cdots = c_k = 0$}
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\definition \textbf{Dimension} $\dim(\mathcal{V}) \in \N$\\
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\smalltext{Max. number of linearly independent vectors in $\mathcal{V}$}
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\definition \textbf{Span} $\text{span}(v_1,\ldots,v_k) := \Bigl\{ \sum_{i=1}^{k}c_iv_i \sep c_i \in \R \Bigr\}$\\
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\subtext{This is a linear subspace of $\{v_1,\ldots,v_k\}$}
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