Why Does Including the Zero Vector Make a Set Linearly Dependent?

[dracolnyte]

Homework Statement

Prove that any collection of vectors which includes $$\theta$$ (zero vector or null vector)is linearly dependent. Thus, null vector cannot be contained in a basis.

The Attempt at a Solution

Well, I know that in order for a collection of vectors to to be linearly dependent, one vector can be expressed as a linear combination of other vectors such as:
let s be some non-zero scalar
let v be vectors

s1v1 + s2v2 + ... + skvk = 0

but let's say that v2 was a zero vector (is this what the question is asking?),
-s2v2 = s1v1 + s3v3 + ... + skvk?
I don't quite get the phrase "any collection of vectors which includes 0(theta)"

 

[Office_Shredder]

Start off by proving the set containing ONLY the zero vector is linearly dependent. It's easiest to use the direct definition here: A set of vectors v₁,..., v_n is linearly dependent if and only if there exist coefficients s<sub>1,..., sn such that not all the si are zero (but some of them are allowed to be) and that

s1v1 + ... + snvn = 0

If you only have the zero vector, what s1 can you pick to satisfy this? Then consider if you add additional vectors... what coefficients can you pick for them?</sub>

 

[dracolnyte]

Okay thanks, I got that one. How about this one?

1. Homework Statement
Prove that if the vectors b1, b2,...bm are linearly dependent, then any collection of vectors which contains the b's must also be linearly dependent

3. The Attempt at a Solution
So to be linearly dependent,
s1b1 + s2b2 +... + smbm = 0
given that the scalar s is not equal to zero.

How should I go about proving this one?

 

[HallsofIvy]

Okay thanks, I got that one. How about this one?

1. Homework Statement
Prove that if the vectors b1, b2,...bm are linearly dependent, then any collection of vectors which contains the b's must also be linearly dependent

3. The Attempt at a Solution
So to be linearly dependent,
s1b1 + s2b2 +... + smbm = 0
given that the scalar s is not equal to zero.

There is NO "scalar s" in what you wrote! You mean that "at least one of the s1, s2, ..., sm is not 0."

How should I go about proving this one?

If a collection of vectors includes the b's, take the coefficients of the addtional vector to all be 0.