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.

  • $\begingroup$ Thank you so much!! $\endgroup$ – Hs P Jun 6 at 12:29

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