Linear Dependency: Proving Vector Independence in V

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In summary, the conversation is about proving the linear independence of three vectors in a vector space V, given that there are already three linearly independent vectors e1, e2, and e3. The conversation also includes a rearranged equation and a hint to use the independence of the original vectors to solve the problem.
  • #1
Yankel
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Hello,

I need some help with this one...any guidance will be appreciated.

In a vector space V there are 3 linearly independent vectors e1,e2,e3. Prove that the vectors:

e1+e2 , e2-e3 , e3+2e1

are also linearly independent.

Thanks...
 
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  • #2
Yankel said:
Hello,

I need some help with this one...any guidance will be appreciated.

In a vector space V there are 3 linearly independent vectors e1,e2,e3. Prove that the vectors:

e1+e2 , e2-e3 , e3+2e1

are also linearly independent.

Thanks...

We want to see when the combination
\[c_1(e_1+e_2)+c_2(e_2-e_3)+c_3(e_3+2e_1)=0\]
Rearranging the terms we get
\[(c_1+2c_3)e_1+(c_1+c_2)e_2+(c_3-c_2)e_3=0\]
Since $e_1,e_2,e_3$ are independent, what does that say about the value of the coefficients $c_1+2c_3, c_1+c_2, c_3-c_2$?

You should have enough information now to finish off this problem.
 

FAQ: Linear Dependency: Proving Vector Independence in V

How do you define linear independence in vector space V?

Linear independence in vector space V refers to a set of vectors that cannot be expressed as a linear combination of each other. In other words, no vector in the set can be written as a combination of the other vectors using scalar multiplication and addition.

What is the process for proving vector independence in V?

To prove vector independence in V, you need to show that the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is when all the coefficients (c1, c2, ..., cn) are equal to 0. This can be done through various methods, such as Gaussian elimination or using determinants.

Can a set of vectors be linearly dependent in one vector space but independent in another?

Yes, a set of vectors can be linearly dependent in one vector space but independent in another. This is because the definition of linear independence depends on the vector space in which the vectors are being considered. A set of vectors may be linearly dependent in one vector space but not in another due to differences in dimensionality or underlying operations.

How does linear dependency affect the span of a set of vectors in V?

Linear dependency affects the span of a set of vectors in V by limiting the number of unique vectors that can be generated through linear combinations of the set. When a set of vectors is linearly dependent, there are redundant vectors that can be expressed as linear combinations of other vectors in the set, resulting in a smaller span.

Can a linearly dependent set of vectors be reduced to a linearly independent set?

Yes, a linearly dependent set of vectors can be reduced to a linearly independent set by removing the redundant vectors. This can be done through Gaussian elimination or other methods such as finding a basis for the set of vectors. The resulting linearly independent set will have the same span as the original set, but with fewer vectors.

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