[Math] Let $u$ and $v$ be nonzero vectors in $R^n$ such that $Au = u$ and $Av = -v$. Prove $u$ and $v$ are linearly independent.

Question

This is my working.

Since $Au = u$ and $Av = -v$, $A$ cannot be a zero vector.

$c_1u + c_2v = 0$

$c_1Au – c_2Av = 0$

$A(c_1 – c_2v) = 0$ Since A is non zero, means $c_1u – c_2v = 0$

$c_1u + c_2v = c_1u – c_2v$

$c_2v = -c_2v$ Since v is non zero, thus $c_2 = 0$

From $c_1u – c_2v = 0$, $c_1u = c_2v$

$c_1u = 0$ Since u is non zero, thus $c_1 = 0$

So since $c_1 = 0$ and $c_2 = 0$, then u and v are linearly independent.

I have tried but is this correct? Now I question myself whether A is nonzero is even correct? Because if that is wrong, everything falls apart. Your advice is appreciated!

A continuation to this is that now $Aw = 0$ is included, and asks to prove that u, v, w are linearly independent. Do I take the same approach?

Answer 1

You can go directly from $c_1u+c_2v=0$ to $A(c_1u+c_2v)=A0=0$ which gives $c_1u-c_2v=0$.