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


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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
	$linear combination of vectors u_1 ,u_2 and u_3 , so v = pu_1 +qu_2 +ru_3 ((( 1)),((−2)),(( 3)) ) = ((p),(p),(p) ) + ((( q)),((2q)),((3q)) ) + ((( 2r)),((−r)),(( r)) ) { ((p+q+2r = 1)),((p+2q−r = −2)),((p+3q+r = 3)) :} or { ((p+q+2r = 1)),(( q−3r = −3)),(( 5r = 10)) :} This unique solution of the triangular system is { ((p = −6)),((q = 3)),((r = 2)) :}. Thus v = −6u_1 +3u_2 +2u_3$
	$l i n e a r c o m b i n a t i o n o f v e c t o r s$ $u_{1}, u_{2} a n d u_{3}, s o v = {p u}_{1} + {q u}_{2} + {r u}_{3}$ $(\begin{matrix} 1 \\ - 2 \\ 3 \end{matrix}) = (\begin{matrix} p \\ p \\ p \end{matrix}) + (\begin{matrix} q \\ 2 q \\ 3 q \end{matrix}) + (\begin{matrix} 2 r \\ - r \\ r \end{matrix})$ ${\begin{cases} p + q + 2 r = 1 \\ p + 2 q - r = - 2 \\ p + 3 q + r = 3 \end{cases} o r {\begin{cases} p + q + 2 r = 1 \\ q - 3 r = - 3 \\ 5 r = 10 \end{cases}$ $T h i s u n i q u e s o l u t i o n o f t h e t r i a n g u l a r$ $s y s t e m i s {\begin{cases} p = - 6 \\ q = 3 \\ r = 2 \end{cases} . T h u s v = - 6 u_{1} + 3 u_{2} + 2 u_{3}$
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