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## Linear Span and Subspaces

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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