Linear Combination: Can 1 0 1 0 be Combined?

In summary, the elements (1 0 1 0, 1 0 0 1, 0 1 0 1, 0 1 1 1) can only be a linear combination of each other if they are dependent, meaning that one can be written as a linear combination of the others. This can be determined by solving the equation (a+ b, c+ d, a+ d, b+ c+ d)= (0, 0, 0, 0) and finding that the only solution is when all constants are 0, meaning the vectors are not dependent. There is no reason to believe that they could be written as linear combinations of each other.
  • #1
ichigo444
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0
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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  • #2
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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  • #3
As HallsofIvy stated, just follow the definition of linear dependence and solve for the constants from there.
 

Related to Linear Combination: Can 1 0 1 0 be Combined?

1. Can 1 0 1 0 be combined using linear combinations?

Yes, 1 0 1 0 can be combined using linear combinations. Linear combinations involve multiplying each number in the set by a constant and then adding them together. In this case, we can multiply 1 by any number and add it to 0 multiplied by another number, resulting in 1 0 1 0.

2. How do you know if a set of numbers can be combined using linear combinations?

A set of numbers can be combined using linear combinations if there exists a linear equation that can be satisfied by those numbers. This means that the numbers can be multiplied by different constants and added together to equal a specific value.

3. Can a linear combination of 1 0 1 0 be written as a single number?

Yes, a linear combination of 1 0 1 0 can be written as a single number. For example, 1 0 1 0 can be written as 10, where 1 is multiplied by 10 and 0 is multiplied by 0, resulting in 10.

4. Is there a limit to the number of terms that can be combined using linear combinations?

No, there is no limit to the number of terms that can be combined using linear combinations. Linear combinations can involve any number of terms, as long as there exists a linear equation that can be satisfied by those terms.

5. How can linear combinations be used in real-life applications?

Linear combinations can be used in various areas of science and engineering, such as physics, chemistry, and economics. They can be used to represent and solve systems of equations, model real-life situations, and make predictions. For example, in chemistry, linear combinations can be used to calculate the concentration of a solution or the amount of reactants needed to produce a certain amount of product.

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