Question

If u, v, and w are nonzero vectors in r 2 , is w a linear combination of u and v?

Answer 1

Not necessarily. $\mathbf u$ and $\mathbf v$ may be linearly dependent, so that their span forms a subspace of $\mathbb R^2$ that does not contain every vector in $\mathbb R^2$.

For example, we could have $\mathbf u=(0,1)$ and $\mathbf v=(0,-1)$. Any vector $\mathbf w$ of the form $(r,0)$, where $r\neq0$, is impossible to obtain as a linear combination of these $\mathbf u$ and $\mathbf v$, since

$c_1\mathbf u+c_2\mathbf v=(0,c_1)+(0,-c_2)=(0,c_1-c_2)\neq(r,0)$

unless $r=0$ and $c_1=c_2$.