Linearly Independent Vectors: Exploring Deletion of Vector xk

In summary, if x1, x2,...,xk are linearly independent vectors in a vector space V, removing one vector from the collection will still result in a linearly independent collection of vectors. This can be proven by showing that the coefficients of the remaining vectors will still be 0, satisfying the definition of linearly independent. This can also be proven using a proof by contradiction, which states that if the remaining coefficients were not 0, it would contradict the definition of linearly independent. Therefore, removing a vector does not affect the linear independence of the collection.
  • #1
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Let x1, x2,...,xk be linear independent vectors in a vector space V.

If we delete a vector, say xk, from the collection, will we still have a linearly independent collection of vectors? Explain.

By deleting a vector from linearly independent span, the other vectors, I believe, will remain independent; however, I don't know how to prove it.
 
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  • #2
Why don't you start with the definition of linearly independent? That's usually a good strategy.
 
  • #3
Well vectors are lin. ind. if the the det doesn't 0 and if all coefficients are 0. Since I know they are already ind., deleting one shouldn't change the coefficients but I don't know how to set it up in a proof still.
 
  • #4
det of what? I don't think that has much to do with the definition of linearly independent. Does it? I suggest you look it up. State it clearly.
 
  • #5
Determinant of the vectors in the span doesn't equal 0 then they are linearly ind.
 
  • #6
Baloney. Determinant is only defined for a square matrix. That's a special case. There's a much more general definition of linear independence.
 
  • #7
What does it mean if x1, x2, x3 are linearly independent? It means that the solution to a1x1 + a2x2 + a3x3 = 0 is ai = 0 for all i=1,2,3. Apply this definition to k vectors.

Now, does this still hold if you take out some vector in {x1,..., xk}? Remove some xi from the set and construct the equation I did above. Does it follow that all the ai's are 0?
 
  • #8
If you remove a vector, the other coefficients should still remain the same = 0.
 
  • #9
Yes, and therefore what?

VeeEight is suggesting a proof by contradiction.
 

1. What does it mean for a set of vectors to be linearly independent?

Linear independence means that no vector in the set can be written as a linear combination of the other vectors in the set. In other words, each vector in the set contributes unique information and cannot be represented by a combination of the other vectors.

2. How do you determine if a set of vectors is linearly independent?

To determine if a set of vectors is linearly independent, you can use the method of Gaussian elimination. Put the vectors into a matrix and perform row operations until the matrix is in reduced row-echelon form. If the matrix has a pivot in every column, then the vectors are linearly independent. If there is a column without a pivot, then the vectors are linearly dependent.

3. What is the purpose of exploring deletion of vector xk in a set of linearly independent vectors?

Exploring the deletion of vector xk in a set of linearly independent vectors allows us to understand how the remaining vectors are affected when one vector is removed. This can help us determine if the remaining vectors are still linearly independent or if they become linearly dependent.

4. Can a set of linearly independent vectors become linearly dependent if a vector is removed?

Yes, it is possible for a set of linearly independent vectors to become linearly dependent if a vector is removed. This can happen if the removed vector was a linear combination of the other vectors in the set, or if the remaining vectors were already linearly dependent and the removal of the vector revealed this dependence.

5. How can exploring deletion of vector xk be applied in real-life situations?

In real-life situations, exploring deletion of vector xk can help us understand the relationships between different variables or components. For example, in economics, deleting a variable from a set of linearly independent variables can help us understand its impact on the overall system. In chemistry, exploring the deletion of a reactant in a chemical reaction can reveal how the remaining reactants are affected and if the reaction can still occur.

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