Write-the-vector-v-1-2-3-as-a-linear-combination-of-vectors-u-1-1-1-1-u-2-1-2-3-and-u-3-2-1-1-

Question Number 118411 by bramlexs22 last updated on 17/Oct/20

W r i t e t h e v e c t o r v = (1, - 2, 3) a s a

l i n e a r c o m b i n a t i o n o f v e c t o r s

u_{1} = (1, 1, 1), u_{2} = (1, 2, 3) a n d u_{3} = (2, - 1, 1)

Answered by benjo_mathlover last updated on 17/Oct/20

l i n e a r c o m b i n a t i o n o f v e c t o r s

u_{1}, u_{2} a n d u_{3}, s o v = {p u}_{1} + {q u}_{2} + {r u}_{3}

(\begin{matrix} 1 \\ - 2 \\ 3 \end{matrix}) = (\begin{matrix} p \\ p \\ p \end{matrix}) + (\begin{matrix} q \\ 2 q \\ 3 q \end{matrix}) + (\begin{matrix} 2 r \\ - r \\ r \end{matrix})

{\begin{cases} p + q + 2 r = 1 \\ p + 2 q - r = - 2 \\ p + 3 q + r = 3 \end{cases} o r {\begin{cases} p + q + 2 r = 1 \\ q - 3 r = - 3 \\ 5 r = 10 \end{cases}

T h i s u n i q u e s o l u t i o n o f t h e t r i a n g u l a r

s y s t e m i s {\begin{cases} p = - 6 \\ q = 3 \\ r = 2 \end{cases} . T h u s v = - 6 u_{1} + 3 u_{2} + 2 u_{3}