Linear Combinations: Will Two Always Produce b=(0,1)?

In summary, there may not always be two different combinations that produce (0,1) of three vectors in R^2. This can happen if the vectors are all parallel and not parallel to (0,1). However, if there is at least one combination that produces (0,1), then there are an infinite number of solutions.
  • #1
porschedude
11
0
Will there always be two different combinations that produce b=(0,1) of three vectors: u, v, and w?

I'm pretty certain that the answer is no, but am I right in saying that with three vectors, assuming they are not all parallel, will always have at least one combination that produces (0,1)
 
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  • #2
You've got that right. If you have three vectors in R^2 then either there are no combinations that produce (0,1) (if they are all parallel and not parallel to (0,1)) or there are an infinite number.
 
  • #3
How can there be an infinite number?
 
  • #4
porschedude said:
How can there be an infinite number?

Say u=(0,1), v=(1,0) and w=(1,1). There are an infinite number of solutions to the equation a*u+b*v+c*w=(0,1). You should be able to show that. It's maybe a little harder to show that if there is one solution, then there are an infinite number.
 

1. How do you define a linear combination?

A linear combination is a mathematical expression in which two or more quantities are multiplied by constants and then added together. The constants are called coefficients, and they determine the weights of each quantity in the combination.

2. Can two linear combinations always produce a specific vector b=(0,1)?

No, two linear combinations may not always result in a specific vector b=(0,1). It depends on the coefficients used in the combinations and the properties of the vectors involved.

3. Under what conditions will two linear combinations produce b=(0,1)?

For two linear combinations to produce b=(0,1), the coefficients must be carefully chosen so that the resulting vector has a magnitude of 1 and a direction of 90 degrees or π/2 radians from the origin.

4. What is the importance of linear combinations in scientific research?

Linear combinations are important in scientific research because they allow us to express complex relationships between variables in a simple and concise manner. They are also used in mathematical models and simulations to understand and predict real-world phenomena.

5. Can linear combinations be extended to more than two quantities?

Yes, linear combinations can involve any number of quantities, not just two. In fact, in linear algebra, linear combinations are often used to represent systems of equations with multiple variables and to find solutions to these systems.

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