Forming a Basis in R^3 - Explain the Correct Solution

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In summary: Therefore, S is not linearly independent and cannot form a basis for R3. In summary, S = {(1,1,1), (-2,1,1), (-1,2,2)} is not a basis for R3 because it is not linearly independent.
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Chadlee88
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i have a question I am trying to work but I am not sure how to do it. I'm given 4 dirrerent answers to choose from (i won't post them because i want to try them myself)

Only one of the following 4 sets of vectors forms a basis of R3.
Explain which one is, and why, and explain why each of the other sets do not form a
basis.


S = {(1,1,1), (-2,1,1), (-1,2,2)}

This one is not because it cannot be expressed as a linear combination right??
 
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S is not a basis for R^3 because it is not linearly independent.
 
  • #3
Chadlee88 said:
i have a question I am trying to work but I am not sure how to do it. I'm given 4 dirrerent answers to choose from (i won't post them because i want to try them myself)

Only one of the following 4 sets of vectors forms a basis of R3.
Explain which one is, and why, and explain why each of the other sets do not form a
basis.


S = {(1,1,1), (-2,1,1), (-1,2,2)}

This one is not because it cannot be expressed as a linear combination right??
Because what "cannot be expressed as a linear combination"?
Grammatically, the "it" in your sentence must refer to "this one", meaning the set of vectors- but it doesn't make sense to talk about expressing a set of vectors as a linear combination of anything.

It is true that S is not a basis for R3 because one of the vectors in S can be expressed as a linear combination of the other two. For example, (1, 1, 1)= -1(-2, 1, 1)+ 1(-1, 2, 2).
 

1. What is a basis in R^3?

A basis in R^3 is a set of three linearly independent vectors that span the entire three-dimensional space. This means that any vector in R^3 can be written as a linear combination of these three basis vectors.

2. Why is it important to have a basis in R^3?

Having a basis in R^3 allows us to easily represent and manipulate vectors in three-dimensional space. It also helps us to understand the structure and properties of three-dimensional objects and systems.

3. How do you determine if a set of vectors in R^3 forms a basis?

To determine if a set of vectors in R^3 forms a basis, we need to check if the vectors are linearly independent and span the entire space. This can be done by setting up a system of equations and solving for the coefficients of the linear combination.

4. What is the correct solution for forming a basis in R^3?

The correct solution for forming a basis in R^3 is to find three linearly independent vectors that span the entire space. This can be done by solving a system of equations or using other methods such as Gram-Schmidt orthogonalization.

5. Can there be more than one basis in R^3?

Yes, there can be infinitely many bases in R^3. This is because there are infinitely many sets of three linearly independent vectors that can span the three-dimensional space. However, any two bases in R^3 will have the same number of vectors, which is three.

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