Mathematics
Kgirl9
2015-11-04 00:10:42
If u, v, and w are nonzero vectors in r 2 , is w a linear combination of u and v?
ANSWERS
Boys
2015-11-04 01:16:15

Not necessarily. [latex]mathbf u[/latex] and [latex]mathbf v[/latex] may be linearly dependent, so that their span forms a subspace of [latex]mathbb R^2[/latex] that does not contain every vector in [latex]mathbb R^2[/latex]. For example, we could have [latex]mathbf u=(0,1)[/latex] and [latex]mathbf v=(0,-1)[/latex]. Any vector [latex]mathbf w[/latex] of the form [latex](r,0)[/latex], where [latex]r eq0[/latex], is impossible to obtain as a linear combination of these [latex]mathbf u[/latex] and [latex]mathbf v[/latex], since [latex]c_1mathbf u+c_2mathbf v=(0,c_1)+(0,-c_2)=(0,c_1-c_2) eq(r,0)[/latex] unless [latex]r=0[/latex] and [latex]c_1=c_2[/latex].

ADD ANSWER