Answer to Question #143191 in Linear Algebra for Ojugbele Daniel

Question

Write the vector v=( 1, -2, 5) as a linear combination of the vector e1=( 1, 1, 1), e2= ( 1, 2, 3) and e3=( 2, -1, 1)

Answer 1

linear combimation has the form:

$v ⃗=α*(e1) ⃗+β*(e2) ⃗+γ*(e3) ⃗$

$(1; -2; 5)=α*(1;1;1)+β*(1;2;3)+γ*(2; -1;1)$

$(1; -2; 5)=(α;α;α;)+(β;2β;3β)+(2γ;-γ;γ)$

find $\alpha, \beta, \gamma:$

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):

$-\beta+3\gamma=3;$

$\beta=3\gamma-3;$ (4)

equation (2) plus equation (3):

$2\alpha+5\beta=3;$

$α=(3-5β)/2;$

$\alpha=(3-5(3\gamma-3))/2;$ (5)

equation (1):

$(3−5(3γ−3))/2+3\gamma-3+2\gamma=1;$

$3-15\gamma+15+10\gamma-6=2;$

$-5\gamma=-10;$

$\gamma=2;$

equation (4):

$\beta=3*2-3;$

$\beta=3;$

equation (1):

$\alpha=1-2\gamma-\beta;$

$\alpha=1-2*2-3;$

$\alpha=-6;$

$v ⃗=-6*(e1) ⃗+3*(e2) ⃗+2*(e3);$

$(1; -2; 5)=-6*(1;1;1)+3*(1;2;3)+2*(2; -1;1);$