Sum of the vector values within a set.
$A+B=\{\vec{x}:\vec{x}=\vec{a}+\vec{b}\text{ for some }\vec{a}∈A \text{ and } \vec{b} ∈ B\}$
$A+(B∪C)=(A+B)∪(A+C)$
$C=\{\vec{x}∈ℝ^2:||\vec{x}||=1\}$
$X=C+\{\vec{e}_2\}$
$Y=C+\{3\vec{e}_1,\vec{e}_2\}$
The span of a set of vectors V is the set of all linear combinations of vectors in V:
$\text{span } V=\{\vec{v}:\vec{v}=a_1\vec{v}_1+\dots+a_n\vec{v}_n \text{ for some } \vec{v}_1, \dots \vec{v}_n ∈ V \text{ and scalars } a_1, \dots a_n\}$
$\text{span} \{\vec{u},\vec{v}\}=\{\vec{x}:\vec{x}=a\vec{u}+b\vec{v} \text{ for some } a,b ∈ ℝ\}$
Line: $\text{span}\{\vec{d}\}+\{\vec{p}\}$ is equivalent to $\vec{x}=t\vec{d}+\vec{p}$
Plane: $\text{span}\{\vec{d}_1, \vec{d}_2\} + \{\vec{p}\}$ is equivalent to $\vec{x}=t\vec{d_1}+s\vec{d_2}+\vec{p}$
(Adding $\{\vec{p}\}$ at the end makes it a Translated Span)
(Regular span can only represent lines/planes passing through the origin)
Span of an empty set:
$\text{span}\{\}=\{\vec{0}\}$
Span of a set of vectors that form a consistent linear equation:
$\text{span}\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}\} = ℝ^2$
Span of multiple vectors with the same or the exact opposite direction:
$\text{span}\{\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 4 \end{bmatrix}\} = \text{span}\{\begin{bmatrix} 1 \\ 2 \end{bmatrix}\}$
Linearly Dependent: A set that contains vectors with the same or exactly opposite direction.