Prove the linear independence of these two vectors

Question

Assume u and v are independent vectors. Using the definition of linear independence, I need should that the vectors a = 4u + 5v and b = 6u - 2v are also linearly independent. I know that I need to show that 2 constants p and q must be equal to 0 in the equation pa + qb = 0, but how is that done using the independence of u and v? What is it about u and v that stops a and b from being scalar multiples of each other?

Answer 1

Assume $p\mathbf a+q\mathbf b=\mathbf 0$. Then $(4p+6q)\mathbf u+(5p-2q)\mathbf v=\mathbf 0$, hence $4p+6q=0$ and $5p-2q=0$. You can conclude from this that $p=q=0$. For example, $19p=(4p+6q)+3\cdot(5p-2q)=0$.