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.
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