Solution:
To solve it show that an arbitrary vectorsl v=(a,b,c) ∈ R3 is a linear combination of u1,u2 and u3 .
Let
v=xu1+yu2+zu3
Hence,
(a,b,c)=x(1,1,1)+y(1,2,3)+z(1,5,8)
Resolve as
x+y+z=a
x+2y+5z=b
x+3y+8z=c
Form the equivalent matrix form of the equations
⎣⎡111123158∣∣∣abc⎦⎤
Reduced echelon form is
⎣⎡10011011−1∣∣∣ab−aa−2b+c⎦⎤
From the reduced echelon form,
−z=a−2b+c
z=−a+2b−c
y+4z=b−a
y=b−a−4z
By substitution,
y=b−a−4(−a+2b−c)
y=3a−7b+4c
Also from the echelon form,
x+y+z=a
x=a−y−z
By substitution,
x=a−(3a−7b+4c)−(−a+2b−c)
x=−a+5b−3c
Therefore,
(a,b,c)=(−a+5b−3c)(1,1,1)+(3a−7b+4c)(1,2,3)+(−a+2b−c)(1,5,8)
Therefore,
u1,u2,u3 span R3
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