Definition. A linear combination of vectors \(\overrightarrow{v_1},\overrightarrow{v_2},\ldots,\overrightarrow{v_k}\) of a vector space \(V\) is a sum of their scalar multiples, i.e., \[c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}+\cdots+c_k\overrightarrow{v_k}\] for some scalars \(c_1,c_2,\ldots,c_k\). The set of all linear combinations of a nonempty set \(S\) of vectors of \(V\) is called the linear span or span of \(S\), denoted by \(\operatorname{Span}(S)\) or \(\operatorname{Span} S\), i.e., \[\operatorname{Span}\{\overrightarrow{v_1},\overrightarrow{v_2},\ldots,\overrightarrow{v_k}\} = \{c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}+\cdots+c_k\overrightarrow{v_k}\;|\; c_1,c_2,\ldots,c_k \text{ are scalars}\}.\] We define \(\operatorname{Span} \varnothing=\{\overrightarrow{0}\}\). When \(\operatorname{Span}\{\overrightarrow{v_1},\ldots,\overrightarrow{v_k}\} =V\), we say \(\{\overrightarrow{v_1},\ldots,\overrightarrow{v_k}\}\) spans \(V\).

Example.

1. \(\operatorname{Span}\{\overrightarrow{e_1},\overrightarrow{e_2},\ldots,\overrightarrow{e_n}\}=\mathbb R^n\).


2. \(\operatorname{Span}\{\overrightarrow{1},\overrightarrow{t},\overrightarrow{t^2},\ldots,\overrightarrow{t^n}\}=P_n\).


3. \(\operatorname{Span}\{\overrightarrow{e_1},\overrightarrow{e_2},\ldots,\overrightarrow{e_n},\ldots\}=\mathbb{R}^{\infty}\) where \(\overrightarrow{e_i}\) is the infinite sequence with \(1\) in the \(i\)th place and \(0\) elsewhere.


4. \(\operatorname{Span}(B)=M_{m, n}(\mathbb R)\) for \(B=\{\overrightarrow{E_{i,j}}\;|\; 1 \leq i \leq m,1 \leq j \leq n\}\) where \(\overrightarrow{E_{i,j}}\) is the \(m\times n\) matrix with the \((i,j)\)-entry 1 and \(0\) elsewhere.

Definition. A subspace of a vector space \(V\) is a nonempty subset \(S\) of \(V\) that satisfies three properties:

1. \(\overrightarrow{0}\) is in \(S\).


2. \(\overrightarrow{u}+\overrightarrow{v}\) is in \(S\) for all \(\overrightarrow{u},\; \overrightarrow{v}\) in \(S\).


3. \(c\overrightarrow{u}\) is in \(S\) for all \(\overrightarrow{u}\) in \(S\) and all scalars \(c\).

In short, a subspace of \(V\) is a nonempty subset \(S\) of \(V\) that is closed under linear combination of vectors, i.e., \(c\overrightarrow{u}+d\overrightarrow{v}\) is in \(S\) for all \(\overrightarrow{u},\; \overrightarrow{v}\) in \(S\) and all scalars \(c,d\). When \(S\) is a subspace of \(V\), we sometimes denote it by \(S\leq V\).

Example.

1. \(\{\overrightarrow{0_V}\}\leq V\) and \(V\leq V\), i.e., \(\{\overrightarrow{0_V}\}\) and \(V\) are subspaces of the vector space \(V\).


2. If \(F\) is the vector space of all real-valued functions, then \(P_n\) is a a subspace of the vector space \(F\).


3. Let \(H\) be the set of all polynomials \(\overrightarrow{p}\) in \(P_n\) such \(\overrightarrow{p}(0)=0\). Note that \[\begin{align*} H&=\{\overrightarrow{p} \in P_n\; |\; \overrightarrow{p}(0)=0\}\\ &=\{a_1t+a_2t^2+\cdots+a_nt^n\; |\; a_1,\ldots,a_n \in \mathbb R\}. \end{align*}\] Then \(H\) is a subspace of the vector space \(P_n\) and consequently a subspace of the vector space \(F\).


4. Let \(H=\left\{ \left[\begin{array}{c}x_1\\x_2\\0 \end{array} \right]\;|\; x_1,x_2\in\mathbb R\right\}\). \(H\) is not a subspace of the vector space \(\mathbb R^2\) but \(H\) is a subspace of the vector space \(\mathbb R^3\).


5. If \(v_1\ldots,v_k\) are vectors of a real vector space \(V\), then \[\operatorname{Span}\{v_1\ldots,v_k\}=\{c_1v_1+\cdots+c_kv_k\; |\; c_1,\ldots,c_k \in \mathbb R \}\] is a subspace of \(V\).

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