Linearly independent vectors and span

[jasoqueso]

So the question is...

Let v1, v2,...,,vn be linearly independent vectors in a vector space V. Show that v2,...,vn cannot span V.

I honestly have found myself completely lost lately and I suck at writing proofs.

So this is what I see,

v1, v2,..., vn is linearly independent iff c1v1+c2v2+ ... + cnvn = 0 and c1=c2=...=cn= 0

however I don't see why taking away v1 would make it not span V anymore.

 

[Tac-Tics]

So the question is...

Let v1, v2,...,,vn be linearly independent vectors in a vector space V. Show that v2,...,vn cannot span V.

I honestly have found myself completely lost lately and I suck at writing proofs.

So this is what I see,

v1, v2,..., vn is linearly independent iff c1v1+c2v2+ ... + cnvn = 0 and c1=c2=...=cn= 0

however I don't see why taking away v1 would make it not span V anymore.

Consider this. How do you write v1 as a linear combination of v2, ..., vn?

Or for a concrete example, in R^3. Take three vectors at right angles called x, y, and z. Can you write x as a linear combination of y and z? Obviously not, but go back to your definition of "span" and "linearly independent" and figure out why!

 

[cup]

Do the following:

1. Re-state the definition of linear independence carefully. (Your language was unclear.)
2. Assume that you can express v₁ as a linear combination of the other vectors.
3. Show that this will contradict linear independence.
4. The conclusion is that you cannot express v₁ using the other vectors.

It's very important to know your definitions precisely!

 

[JG89]

In an n-dimensional vector space, a linearly independent set of n vectors is a basis for that space. Also, in an n-dimensional vector space, every basis for that space has the same amount of vectors. A basis is a linearly independent set that spans the space. This makes the rest of the proof easy.