Find all scalars c_{1} , c_{2}, c_{3} such that c_{1}(1 , -1, 0) + c_{2}(4, 5, 1) + c_{3}(0, 1, 5) = (3, 2, -19)



Answered question


Find all scalars c1,c2,c3 such that c1(1,1,0)+c2(4,5,1)+c3(0,1,5)=(3,2,19)

Answer & Explanation

Nathanael Webber

Nathanael Webber

Skilled2021-03-12Added 117 answers

The equation c1(1,1,0)+c2(4,5,1)+c3(0,1,5)=(3,2,19)
becomes (c1c1,0)+(4c2,5c2,c2)+(0,c35c3)=(3,2,19)
which can be written as (c1+4c2,c2+5c2+c3,c2+5c3)=(3,2,19)
This gives us the the system c1+4c2=3
From the first equation we get c1=34c2
From the third equation we get c2=195c3
Thus, c1=34c2=34(195c3)+c3=244c3174=2
This yields 44c3=176c3=4
Thus, c2=195c3=195(4)c2=1
and c2=79+20c3=79+20(4)c1=1
Therefore, to conclusude, c1=1,c2=1,c3=4

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