Can Deleting a Vector from Linearly Independent Vectors Preserve Independence?

[Dustinsfl]

Let x₁, x₂,...,x_k be linear independent vectors in a vector space V.

If we delete a vector, say x_k, 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.

 

[Dick]

Why don't you start with the definition of linearly independent? That's usually a good strategy.

 

[Dustinsfl]

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.

 

[Dick]

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.

 

[Dustinsfl]

Determinant of the vectors in the span doesn't equal 0 then they are linearly ind.

 

[Dick]

Baloney. Determinant is only defined for a square matrix. That's a special case. There's a much more general definition of linear independence.

 

[VeeEight]

What does it mean if x₁, x₂, x₃ are linearly independent? It means that the solution to a₁x₁ + a₂x₂ + a₃x₃ = 0 is a_i = 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,..., x_k}? Remove some x_i from the set and construct the equation I did above. Does it follow that all the a_i's are 0?

 

[Dustinsfl]

If you remove a vector, the other coefficients should still remain the same = 0.

 

[HallsofIvy]

Yes, and therefore what?

VeeEight is suggesting a proof by contradiction.