Linear Combination, Span
Definition: Linear Combination1
Let $\mathbf{w}$ be a vector in the vector space $V$. If $\mathbf{w}$ can be expressed as follows for vectors $\mathbf{v}_{1},\mathbf{v}_{2},\cdots ,\mathbf{v}_{r}$ in $V$ and arbitrary constants $k_{1}, k_{2}, \cdots, k_{r}$, then $\mathbf{w}$ is called a linear combination of $\mathbf{v}_{1},\mathbf{v}_{2},\cdots ,\mathbf{v}_{r}$.
$$ \mathbf{w} = k_{1}\mathbf{v}_{1} + k_{2}\mathbf{v}_{2} + \cdots + k_{r}\mathbf{v}_{r} $$
Additionally, in this case, the constants $k_{1}, k_{2}, \cdots, k_{r}$ are referred to as the coefficients of the linear combination $\mathbf{w}$.
Explanation
Though it might seem unfamiliar presented in a formulaic manner, it’s not a complex concept. The representation of vectors in a two-dimensional Cartesian coordinate system is precisely the linear combination of two unit vectors $\hat{\mathbf{x}} = (1,0)$ and $\hat{\mathbf{y}} = (0,1)$.
$$ \mathbf{v} = (v_{1}, v_{2}) = (v_{1},0)+(0,v_{2}) = v_{1}(1,0) + v_{2}(0,1) = v_{1}\hat{\mathbf{x}} + v_{2} \hat{\mathbf{y}} $$
Theorem
Let $S = \left\{ \mathbf{w}_{1}, \mathbf{w}_{2}, \dots, \mathbf{w}_{r} \right\}$ be a non-empty subset of the vector space $V$. Then the following hold.
(a) Let $W$ be the set of all possible linear combinations of elements of $S$. $W$ is a subspace of $V$.
(b) The $W$ from (a) is the smallest subspace of $V$ that includes $S$. That is, if $W^{\prime}$ is a subspace of $V$ that includes $S$, then the following equation holds.
$$ S \subset W \le W^{\prime} $$
Proof
(a)
To demonstrate that $W$ is closed under addition and scalar multiplication, one can apply the subspace test as follows.
$$ \mathbf{u} = c_{1} \mathbf{w}_{1} + c_{2} \mathbf{w}_{2} + \cdots + c_{r} \mathbf{w}_{r}, \quad \mathbf{v} = k_{1} \mathbf{w}_{1} + k_{2} \mathbf{w}_{2} + \cdots + k_{r} \mathbf{w}_{r} $$
(A1)
$\mathbf{u}+\mathbf{v}$ is as follows.
$$ \mathbf{u} +\mathbf{v} = ( c_{1} + k_{1} ) \mathbf{w}_{1} + ( c_{2} + k_{2} ) \mathbf{w}_{2} + \cdots + ( c_{r} + k_{r} ) \mathbf{w}_{r} $$
Since this is a linear combination of $\mathbf{w}_{1}, \mathbf{w}_{2}, \dots, \mathbf{w}_{r}$, $\mathbf{u} + \mathbf{v} \in W$ is true.
(M1)
For any constant $k$, $k\mathbf{u}$ is as follows.
$$ k\mathbf{u} = ( k c_{1} ) \mathbf{w}_{1} + ( k c_{2} ) \mathbf{w}_{2} + \cdots + ( k c_{r} ) \mathbf{w}_{r} $$
Since this is a linear combination of $\mathbf{w}_{1}, \mathbf{w}_{2}, \dots, \mathbf{w}_{r}$, $k\mathbf{u} \in W$ is true.
Conclusion
Since $W$ is closed under addition and scalar multiplication, by the subspace test, $W$ is a subspace of $V$.
$$ W \le V $$
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(b)
Assuming $W^{\prime}$ is a subspace of $V$ that includes $S$, since $W^{\prime}$ is closed under addition and scalar multiplication, all linear combinations of elements of $S$ are elements of $W^{\prime}$. Therefore,
$$ W \le W^{\prime} $$
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Definition: Span
The $W$ in the theorem is referred to as the subspace of $V$ spanned by $S$. Furthermore, it is said that the vectors $\mathbf{w}_{1}, \mathbf{w}_{2}, \dots, \mathbf{w}_{r}$ span $W$, which is denoted as follows.
$$ W = \text{span}\left\{ \mathbf{w}_{1}, \mathbf{w}_{2}, \dots, \mathbf{w}_{r} \right\} \quad \text{or} \quad W = \text{span}(S) $$
Explanation
The concept of spanning is necessary to contemplate the smallest set that contains certain elements. Indeed, the above theorem highlights this point. Additionally, eliminating all redundant elements from $S$ itself would make it the basis of a vector space.
Theorem
Let $S = \left\{ \mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{r} \right\}$ and $S^{\prime} = \left\{ \mathbf{w}_{1}, \mathbf{w}_{2}, \dots, \mathbf{w}_{r} \right\}$ be non-empty subsets of the vector space $V$. Then,
$$ \text{span} \left\{ \mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{r} \right\} = \text{span} \left\{ \mathbf{w}_{1}, \mathbf{w}_{2}, \dots, \mathbf{w}_{r} \right\} $$
The necessary and sufficient condition for this to hold is that all vectors of $S$ can be expressed as linear combinations of vectors of $S^{\prime}$, and all vectors of $S^{\prime}$ can be expressed as linear combinations of vectors of $S$.
Howard Anton, Elementary Linear Algebra: Applications Version (12th Edition, 2019), p220-222 ↩︎