# Two vectors are linearly independent?

Let $$x, y, z$$ be vectors in vector space $$V$$. Suppose $$z \notin L(x,y)$$ , where $$L(x,y)$$ is the linear span of $$x, y$$.

Show that $$x, y$$ are linearly independent iff x+z, y+z are linearly independent.

I can easily show that $$x, y$$ are linearly independent implies linear independence of $$x+z, y+z$$. But I have a trouble with showing converse!. I want someone who help me~~

The converse is not true. Let $$x=(1,0)$$, $$y=(2,0)$$ and $$z=(0,1)$$. Then $$x$$ and $$y$$ are not linearly independent, but $$x+z=(1,1)$$ and $$y+z=(2,1)$$ are.

• Thank you so much!! – Hs P Jun 6 at 12:29