Linear independece

[white pollen]

A set S of vectors in a vector space is said to be linearly indepedent if:
a1 * x1 + a2 * x2 + ... + an * xn = 0 ===> a1 = a2 = ... = an = 0

I see a proposition in my lecturer notes that says: "If S contains the null vector, 0, then it's linearly DEPEDENT". Why??
We can write the null vector with a linear combination of all the vectors in the set S with 0 scalars and so it should be by definition linearly independent

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[civic blaze]

Because you can multiply it by any constant and it will remain a zero vectors.
So, any collection with it has a nontrivial linear combination resulting in a zero vector.

In other words, the zero vector is linearly dependent with any other vector.

[white pollen]

Hmm

But what if it's a vector space over a field with 1 element

Your argument fails then

Actually

Are there any fields with 1 element?

[civic blaze]

Well, considering the zero vector must be an element of a vector space, then the space consisting of just it is a vector space.

But then there's no point in talking about linear dependence 😄

[white pollen]

I see

Thank you

+close