Linear Span

The linear span (sometimes just called span) of a set of Vectors (i.e. elements of a Vector Space) is the set of all the Linear Combinations derived from that set.

The result of the span operator always takes in a set of elements from a vector space and returns another set of vectors.

$V=\text{span(S)}$ can be expressed in different phrases:

• $S$ spans $V$
• $S$ is the spanning set of $V$
• $V$ is spanned by $S$

Definition §

$$\text{span}(S)=\text{All linear combinations from vectors in S}$$

Let $V_{\mathbb{F}}$ be a vector space. Then

V = \text{span}(S) \iff S \subseteq V \text{ and V can *only* contain linear combinations of S}$$ ### Theorem: Every Span of a Vector Space is a Subspace Let $V_{\mathbb{F}}$ be a vector space and ${\vec{v}_{1}, \vec{v}_{2},..\vec{v}_{n}} \in V$ . Then $W = \text{span}\{\vec{v}_{1}, \vec{v}_{2}, ..\vec{v}_{n} \}$ is a [Subspace](subspace.md) of $V_{\mathbb{F}}$. ### Checking Span via Matrices We can use [matrices](matrix.md) to determine if a set of vectors span a vector space. Let $V_{\mathbb{F}}^n$ be a vector space from the set $V^n$ which consists all $n$-tuples fixed over the field $\mathbb{F}$. For example, $\mathbb{R}_{\mathbb{R}}^2$ is the vector space of all $2$-tuples containing real numbers, fixed over the field of real numbers. #ASK am i putting the vectors $\vec{v},etc$ and $t_{1}$ in the right sets? Let $S$ be the set of vectors from the vector space $V_\mathbb{F}^n$ and $V^n$ be the set $S$ spans. We need to check if:

V^n = \text{span}(S_{k})

Where $S_{k}$ is defined as having vectors:

S_{k} = { \vec{v}{1}, \vec{v}{2}, \vec{v}{3} \dots \vec{v}{k}}| \vec{v}_{1\dots k} \in V

$$Thenwecanrepresenteachvectorasa[columnmatrix](Matrix$$

\boldsymbol{M} = [\boldsymbol{v}{1}, \boldsymbol{v}{2},\dots \boldsymbol{v}_{k}]

= \begin{bmatrix} {\boldsymbol{v}{1}}{1} & {\boldsymbol{v}{2}}{1} & \dots & {\boldsymbol{v}{k}}{1} \ {\boldsymbol{v}{1}}{2} & {\boldsymbol{v}{2}}{2} & \dots & {\boldsymbol{v}{k}}{2} \ \dots & \dots & \dots & \dots \ {\boldsymbol{v}{1}}{n} & {\boldsymbol{v}{2}}{n} & \dots & {\boldsymbol{v}{k}}{n} \end{bmatrix}

$$Ifwe[augment](Matrix$$

\boldsymbol{X}= \begin{bmatrix} t_{1} \ t_{2} \ \dots \ t_{n} \end{bmatrix} , t \in V

$$Thenwegetthematrix$$

[\boldsymbol{M|\boldsymbol{X}}]

This is a [Matrix Representation Of A Linear System](Matrix%20Representation%20Of%20A%20Linear%20System.md), specifically, a non-[homogenous](Linear%20System.md#Homogeneity) system and can be solved via [Gauss-Jordan Elimination](Gauss-Jordan%20Elimination.md) to get the [RREF](Reduced%20Row%20Echelon%20Form.md). If the system is [consistent](Consistency.md) then $S_{k}$ spans $V^n$. This can be checked if: * $\text{rank(}[\boldsymbol{M}|\boldsymbol{X}])= \text{rank}(\boldsymbol{M})$ * $\text{rank}(\boldsymbol{M}) = n$ - Because if this isn't the case, the system of $[\boldsymbol{M|X}]$ contains a row with zeroes in $\boldsymbol{M}$ but not in $\boldsymbol{X}$, making it inconsistent This also implies that $k \geq n$ because the [Rank](Rank.md) is always lesser or equal to $\text{min}(k,n)$. If $k < n$ then $\text{rank}(\boldsymbol{M}) = k$ but $\text{rank}(\boldsymbol{M}|\boldsymbol{X}) = n$, making the system inconsistent

k \geq n \iff \text{There must be at least $n$ or more vectors to span $V^n$}$$for this to occur.

Examples §

The 3D Real Space, ${\mathbb{R}}^3$ §

In the vector space ${\mathbb{R}}^3$ fixed over $\mathbb{F}=\mathbb{R}$, we can obtain the following vector spaces using spans of vectors $\vec{a},\vec{b},\vec{c}\in {\mathbb{R}}^3$ and are all linearly independent

Possible Vector Subspaces	Vectors Required ($k$)	Defined by span
The Zero Vector Space: $\{\vec{0}\}$	1	$\text{span}(\{\vec{a}\})$
A Vector Line	1	$\text{span}(\{\vec{a}\})\Vert \vec{a}\ne \vec{0}$
A Plane	2	$\text{span}(\{\vec{a},\vec{b}\})\Vert \vec{a},\vec{b}\ne \vec{0}$
${\mathbb{R}}^3$	3	$\text{span}(\{\vec{a},\vec{b},\vec{c}\})\Vert \vec{a},\vec{b},\vec{c}\ne \vec{0}$

Any more vectors are irrelevant, because the subspaces above are the only valid subspaces in ${\mathbb{R}}^3$

We can prove this using the Checking if vectors span a vector space