Definition 1.3.1. Let \(A\) be a set equipped with an associative addition operation and an additive identity \(0_A\). For each \(n \in \mathbb{N}_0\) and each function
\[a:\lbrace1,\dots,n\rbrace\longrightarrow A,\]the finite sum of the family \(a_1,\dots,a_n,\) denoted by
\[\sum_{i=1}^{n} a_i,\]is defined recursively as follows:
\[\sum_{i=1}^{0} a_i = 0_A,\]and, for each \(k \in \mathbb{N}_0\),
\[\sum_{i=1}^{k+1} a_i = \left(\sum_{i=1}^{k} a_i\right) + a_{k+1}.\]When \(n = 0\), the domain \(\lbrace 1,\dots,0\rbrace\) is understood to be \(\varnothing\), and \(a:\varnothing \to A\) is the unique empty function. The sum \(\sum_{i=1}^{0} a_i\) is called the empty sum.
Definition 1.3.2. Let \(\mathbb V\) be a vector space over \(\mathbb F\), and let \(S \subseteq \mathbb V\). A vector \(v \in \mathbb V\) is a linear combination of the vectors in \(S\) if
\[\text{there exist }n\in\mathbb N_0,\quad u:\lbrace1,\dots,n\rbrace \to S,\quad a:\lbrace1,\dots,n\rbrace \to\mathbb F\]such that
\[v = \sum_{i=1}^{n} a_i u_i.\] If \(v\) is a linear combination of the vectors in \(S\) with \(n \geq 1,\) we also say that \(v\) is a linear combination of \(u_1, \dots, u_n,\) and we call \(a_1, \dots, a_n\) the coefficients of the linear combination.
Definition 1.3.3. Let \(S\) be a subset of a vector space \(\mathbb V\) over a field \(\mathbb F.\) The span of \(S\) is the set \(\text{span} (S)\) defined by
\[\operatorname{span}(S) = \left\lbrace v \; : \; v\text{ is a linear combination of the vectors in } S \right\rbrace.\]Theorem 1.3.4. Let \(S\) be a subset of a vector space \(\mathbb V\) over a field \(\mathbb F.\) Then \(\text{span} (S)\) is a subspace of \(\mathbb V\) that contains \(S.\) Moreover, any subspace of \(\mathbb V\) that contains \(S\) also contains \(\text{span} (S).\)
Proof.
References
- Friedberg, S. H., Insel, A. J., & Spence, L. E. (2025). Linear algebra (5th ed.). Pearson Education South Asia Pte Ltd.