1. Linear combinations of vectors

a

Proof: Consider any vector $u \in R^{2}$

u = λ v + μ w

(u_{1} u_{2}) = λ (11) + μ (- 1 1)

λ = \frac{u _{1} + u _{2}}{2}, μ = \frac{u _{2} - u _{1}}{2}

we see $λ$ and $μ$ can always be solved in terms of $u_{1}$ and $u_{2}$ q.e.d.

u = 100

suppose $u = λ v + μ w$ , $λ$ must be 1 because of $u_{1}$ , but then $μ$ can never be adjusted so that $u_{2}$ and $u_{3}$ are both 0

x (1585) + (1 - x) (3565) = (2377)

x = \frac{3}{5}, y = 1 - x = \frac{2}{5}

\hat{D} := {(g t) \in R^{2} : g + t = 100, g \geq 0, t \geq 0, 15 \leq g \leq 35, 65 \leq t \leq 85}