Proof with two linearly independent vectors

[Tristan Pouliot]

Hi there,
I have to this as a homework but I'm not sure where to start these proofs..

Let the vector u and v be linearly independent vectors.
A) show that (u, u + v, v) is a set of linearly dependent vectors.
B) show that (u, u + v, 0) is a set of linearly dependent vectors.
C) show that (u + v, u-v) is a set of linearly independent vectors.

 

[pka]

Before we can help. we need to know how your textbook/notes defines linearly independent vectors.

 

[Tristan Pouliot]

Of course,
In my notes it's written that if a set of n vectors having the same number of components has a unique solution (k1v1 + k2v2 + k3v3 + ... + knvn = 0), we say that it is linearly independent. If the set has more than one solution, it is linearly dependent.

 

[pka]

Thank you for the reply. You may want to read this page.
You are given that \(\vec{u}~\&~\vec{v}\) are independent.
Let's look at part C.
Suppose that \(\alpha (\vec{u}+\vec{v})+\beta (\vec{u}-\vec{v})=\vec{0}\)
Given that can you show that \(\alpha\text{ or }\beta\) is zero?

 

[Tristan Pouliot]

Thank you for your help! And I think so