Linear Combination: Can 1 0 1 0 be Combined?

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SUMMARY

The discussion centers on the linear dependence of the vectors (1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 0, 1), and (0, 1, 1, 1). It concludes that these vectors are linearly independent, as the only solution to the equation a(1, 0, 1, 0) + b(1, 0, 0, 1) + c(0, 1, 0, 1) + d(0, 1, 1, 1) = (0, 0, 0, 0) is a = b = c = d = 0. Therefore, none of the vectors can be expressed as a linear combination of the others, confirming their independence.

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  • Understanding of linear algebra concepts, specifically linear combinations and linear dependence.
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  • Study the concept of linear independence in greater detail.
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators teaching these concepts. This discussion is beneficial for anyone looking to deepen their understanding of vector spaces and linear combinations.

ichigo444
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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?
 
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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?
 
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As HallsofIvy stated, just follow the definition of linear dependence and solve for the constants from there.
 

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