Solution SS.M10 Chris Black

1. The span of a single vector $\vect{v}$ is the set of all linear combinations of that vector. Thus, $\spn{\vect{v}} = \setparts{\alpha\vect{v}}{\alpha\in\mathbb{R}}$. This is the line through the origin and containing the (geometric) vector $\vect{v}$. Thus, if $\vect{v}=\colvector{v_1\\v_2}$, then the span of $\vect{v}$ is the line through $(0,0)$ and $(v_1,v_2)$.
2. Two vectors will span the entire plane if they point in different directions, meaning that $\vect{u}$ does not lie on the line through $\vect{v}$ and vice-versa. That is, for vectors $\vect{u}$ and $\vect{v}$ in $\real{2}$, $\spn{\vect{u}, \vect{v}} = \real{2}$ if $\vect{u}$ is not a multiple of $\vect{v}$.