Linear Algebra Proof -- Linear Dependence

[TheWrathOfMath]

Prove or Disprove: If v1, v2, v3 are linearly dependent vectors, then v1+v2, v1+v3, v2+v3 are linearly dependent.

I think that this is a true statement.
I tried proving it but got stuck.
Below is a photo of what I have so far.
I would greatly appreciate your assistance.

 

[Steven G]

Please tell us the definition of a set of linear independent vectors that you use in your class (or textbook).

EDIT: Please do not post your question in the heading!

 

[Steven G]

You never showed that any linear combination ow w1, w2 and w3 is a linear combination of v1, v2 and v3. You never concluded that the vectors v1+V2, V1+V3, V2+V3 are L.D.

A linear combination of v1, v2 and v3 is of the form c1v1 + c2v2 + c3v3, where c1, c2, c3 in F

You seem to have the right idea but your proof is far from complete.

 

[TheWrathOfMath]

You never showed that any linear combination ow w1, w2 and w3 is a linear combination of v1, v2 and v3. You never concluded that the vectors v1+V2, V1+V3, V2+V3 are L.D.

A linear combination of v1, v2 and v3 is of the form c1v1 + c2v2 + c3v3, where c1, c2, c3 in F

You seem to have the right idea but your proof is far from complete.

I fixed it.
Please see the attached file.

Please tell us the definition of a set of linear independent vectors that you use in your class (or textbook).

EDIT: Please do not post your question in the heading!

Linearly dependent -- when there is a nontrivial linear combination of a set of vectors that equals the zero vector.
If no such linear combination exists, then the vectors are linearly independent.

We did not learn basis and dimension yet.

Please tell us the definition of a set of linear independent vectors that you use in your class (or textbook).

EDIT: Please do not post your question in the heading!

I apologize. I fixed it.

You never showed that any linear combination ow w1, w2 and w3 is a linear combination of v1, v2 and v3. You never concluded that the vectors v1+V2, V1+V3, V2+V3 are L.D.

A linear combination of v1, v2 and v3 is of the form c1v1 + c2v2 + c3v3, where c1, c2, c3 in F

You seem to have the right idea but your proof is far from complete.

Just to clarift: When I wrote, "I fixed it," I did not mean that I solved the problem.

I fixed it.
Please see the attached file.

 

[Steven G]

So can you finish the proof?

 

[TheWrathOfMath]

So can you finish the proof?

Is what I did so far alright?

If so, should I simply substitute b1+b3=a1, b1+b3=a2, b2+b3=a3 into the linear combination and get that a1v1+a2v2+a3v3=0?

 

[Steven G]

Is what I did so far alright?

If so, should I simply substitute b1+b3=a1, b1+b3=a2, b2+b3=a3 into the linear combination and get that a1v1+a2v2+a3v3=0?

The proof so far is fine. What do you think you need to add?

 

[TheWrathOfMath]

I need to somehow relate it to the fact that since v1, v2 and v3 are linearly dependent.
In fact, the expression I got at the end matches the definition of linear dependency of vectors v1, v2, v3.

The proof so far is fine. What do you think you need to add?

 

[topsquark]

BUMP

Please do not bump. It is rude. Someone will get to your post.

-Dan

 

[TheWrathOfMath]

Please do not bump. It is rude. Someone will get to your post.

-Dan

I apologize and I will refrain from doing it in the future.