# Why such a vectors are linearly independent? [closed]

Assume that $A$ is an linear operator on a real vector spacev $V$.

I wish to prove that if for some $x,y \in V$ such that $x\neq 0$ or $y \neq 0$ and some $a,b \in \mathbb R$ and $b\neq 0$, the following conditions hold $$Ax=ax-by, Ay=ay+bx$$ then $x,y$ are linearly independent.

## closed as off-topic by Semiclassical, Jyrki Lahtonen, Najib Idrissi, Vedran Šego, beep-boopOct 11 '14 at 14:50

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• What did you try? A proof by contradiction would begin by assuming $x,y$ are not linearly independent. Combine the expressions resulting from that assumption with the conditions above involving $a,b,x,y$ and see if this gives a contradiction. – hardmath Oct 11 '14 at 12:44

Suppose the contrary, i.e., that they are linearly dependent. Then $x=ky$, where $k \ne 0$, so
This is impossible as $V$ is a real vector space.