Linear combinations and spanning

Overview §

A linear combination of the elements of a vector space $V$ defines another element of $V$. The spanning set is the set of all possible linear combinations that can be made out of elements in a subset of a vector space.

Note the following phrasing:

• “The span of $S$” = the vector subspace which is the set of all possible linear combinations we can make out of elements in the set $S$.
• “Spanning set” = the set of vectors which linear combinations are made out of.
• “$S$ spans $V$” = the span of the subset $S\subset V$ is the whole space $V$. That is, everything in $V$ is a linear combination of the elements in $S$.

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Linear combinations §

Linear combinations are a meaningful way to make vectors out of other vectors.

Definition: Linear combination

If $a_1,\cdots ,a_n\in \mathbb{F}$ and $V$ is a vector space over $\mathbb{F}$, the linear combination of $v_1,\cdots ,v_n\in V$, then is an element of $V$ given by

$$a_1{v}_1+\cdots +a_n{v}_n.$$

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Spans §

Definition: Span

If $V$ is a vector space over the field $\mathbb{F}$, and $S\subset V$ is any subset, the span of $S$ is the set of all linear combinations that can be made out of the elements of $S$.

$$\text{span}(S)=\{a_1{v}_1+\cdots +a_n{v}_n\mid a_1,\cdots ,a_n\in \mathbb{F}\text{ and }{v}_1,\cdots ,v_n\in S\}\subset V$$

We can also define spans constructively: the set $v_1,\cdots ,v_n$ spans $V$ if every element of $V$ can be written as a linear combination of the set of vectors.

$$\text{span}(v_1,...,v_n)={\forall }_{v\in V}{\exists }_{a_1,\cdots ,a_n\in \mathbb{F}}[a_1{v}_1+\cdots +a_n{v}_n]$$

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Spanning subspaces §

• (Proposition 25) See notes for example. The set of eventually-zero functions $S=\{f_0,f_1,f_2\cdots \}\subset \text{Map}(\mathbb{N},\mathbb{F})$ is spanned by $S$.

$$span(S)={\text{Map}}_{fin}$$

• (Proposition 26) If $S\subset V$ is an arbitrary subset of vector space $V$, the set $\text{span}(S)$ satisfies the following properties:
  • The set $\text{span}(S)\subset V$ is a linear subspace;
  • We have $S\subset \text{span}(S)$;
  • If $S\subset W\subset V$ for a linear subspace $W$, then we have $\text{span}(S)$ in $W$;
• Intuitively, $\text{span}(S)$ is the smallest subspace containing $S$.

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Review §

Prove the following statements:

• Any ${\mathbb{F}}^n$ space can be described all linear combinations of the standard vectors $e_1,...,e_n$.
• A linear combination of elements of $V$ defines another element of $V$.
• Prop. 26 by induction.
• The line through a vector $L_v=\{av\mid a\in \mathbb{F}\}\subset V$ is just $L_v=\text{span}(V)$.
• The span of the zero vector is just the subspace containing the zero vector $\text{span}(\vec{0})=\{\vec{0}\}$.
• In ${\mathbb{F}}^3$, $\text{span}((1,0,-1),(0,1,-1))=\{(x,y,z)\in {\mathbb{F}}^3\mid x+y+z=0\}$.