\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 $\{v_1,\ldots,v_k\}$}

