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\\eq0`, 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)\\eq(r,0)`

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