Linear Combination: Can 1 0 1 0 be Combined?

[ichigo444]

How can at least one of these elements (1 0 1 0, 1 0 0 1, 0 1 0 1, 0 1 1 1) be a linear combination of the other? Or can it?

 

[HallsofIvy]

Look at a(1, 0, 1, 0)+ b(1, 0, 0, 1)+ c(0, 1, 0, 1)+ d(0, 1, 1, 1)= (0, 0, 0, 0). That will have a solution with at least one of a, b, c, and d not 0 if and only if the vectors are dependent. And that is the only situation in which one can be written as a linear combination of the other.

That equation is the same as (a+ b, c+ d, a+ d, b+ c+ d)= (0, 0, 0, 0) and gives the four equations a+ b= 0, c+ d= 0, a+ d= 0, b+ c+ d= 0. From the second equation, c= -d so b+ c+ d= b- d+ d= b= 0. From the first equation, a+ b= a +0= a = 0. From the third, a+ d= 0+ d= d= 0, and from the fourth 0+ c+ 0= c= 0.

The only solution to that equation is a= b= c= d= 0 so the vectors are not dependent and one cannot be written as a linear combination of the others.

Did you have any reason to think they could?

 

[Anonymous217]

As HallsofIvy stated, just follow the definition of linear dependence and solve for the constants from there.