linear combimation has the form:
v⃗=α∗(e1)⃗+β∗(e2)⃗+γ∗(e3)⃗
(1;−2;5)=α∗(1;1;1)+β∗(1;2;3)+γ∗(2;−1;1)
(1;−2;5)=(α;α;α;)+(β;2β;3β)+(2γ;−γ;γ)
find α,β,γ:
system of equations:
α+β+2γ=1,(1)α+2β−γ=−2,(2)α+3β+γ=5;(3)
solution system:
equation (1) minus equation (2):
−β+3γ=3;
β=3γ−3; (4)
equation (2) plus equation (3):
2α+5β=3;
α=(3−5β)/2;
α=(3−5(3γ−3))/2; (5)
equation (1):
(3−5(3γ−3))/2+3γ−3+2γ=1;
3−15γ+15+10γ−6=2;
−5γ=−10;
γ=2;
equation (4):
β=3∗2−3;
β=3;
equation (1):
α=1−2γ−β;
α=1−2∗2−3;
α=−6;
v⃗=−6∗(e1)⃗+3∗(e2)⃗+2∗(e3);
(1;−2;5)=−6∗(1;1;1)+3∗(1;2;3)+2∗(2;−1;1);
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