# Linearly independent

Dustinsfl
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.

Homework Helper
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.

Homework Helper
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.

Homework Helper
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 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?

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