# Thread: If V is an abstract vector space, which of these sets are linearly independent?

1. ## If V is an abstract vector space, which of these sets are linearly independent?

The main issue I'm having with this question is the term "abstract vector space". It hasn't popped up in our course material, and I'm sure the adjective "abstract" changes the meaning significantly. Here's the question:

Let $\displaystyle V$ be an abstract vector space. If u, v and w are linearly independent vectors in $\displaystyle V$ determine which, if any of the following sets are linearly independent:
$\displaystyle S_1=\{u,u+v,v+w,w\}$
$\displaystyle S_2=\{u,u+v,u-v,v+w\}$
$\displaystyle S_3=\{u,u+v,u-v+w\}$
In each case, justify your conclusions.

I'm under the assumption that none of these sets are linearly independent, since in each case the set has at least one linearly dependent vector (usually dependent on the first vector). But considering the addition of the word "abstract", I have to question whether my assumption is correct.

2. Originally Posted by Runty
The main issue I'm having with this question is the term "abstract vector space". It hasn't popped up in our course material, and I'm sure the adjective "abstract" changes the meaning significantly. Here's the question:

Let $\displaystyle V$ be an abstract vector space. If u, v and w are linearly independent vectors in $\displaystyle V$ determine which, if any of the following sets are linearly independent:
$\displaystyle S_1=\{u,u+v,v+w,w\}$
$\displaystyle S_2=\{u,u+v,u-v,v+w\}$
$\displaystyle S_3=\{u,u+v,u-v+w\}$
In each case, justify your conclusions.

I'm under the assumption that none of these sets are linearly independent, since in each case the set has at least one linearly dependent vector (usually dependent on the first vector). But considering the addition of the word "abstract", I have to question whether my assumption is correct.
Abstract just means that no specifics are given.

3. Originally Posted by Drexel28
Abstract just means that no specifics are given.
Thanks, I was worried it meant something else.

So, am I right answering that none of them are linearly independent sets since not all of their vectors are linearly independent?

4. Since u, v, and w are independent they span (and form a basis for) some three dimensional subspace of V.

Every vector in every set is a linear combination of u, v, and w and so are in that subspace. But there are four vectors in each set, larger than the dimension of the subspace. Therefore, they cannot be independent.