Suppose $u, v, w$ are linearly independent in $V$ and $x \in V$. Prove $u + x, v + x, w + x$ are always linearly independent.
I'm stuck on this problem because I don't know what to do with $x$. Any hints or help would be appreciated.
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Similar question Prove that, if ${u,v,w}$ is a basis for a vector space $V$, then so is ${u+v, v+w, u+v+w}$. – Git Gud Nov 10 '14 at 01:08
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I recommend changing "random" to "arbitrary" in the title. I thought the vectors were going to be random variables (as in probability). – Matt Samuel Nov 10 '14 at 01:15
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You're stuck because the result is false as it is stated now. Let ${\bf x} = -{\bf u}$, or the opposite of any of the other two vectors. If $\{{\bf u},{\bf v}, {\bf w}, {\bf x}\}$ is linearly independent, for example, it seems true, and follows straight from definition. Maybe there is some other hypothesis we can add to fix this.
Ivo Terek
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Thanks you're totally right! the question was asking if the statement was true or false. – user184679 Nov 10 '14 at 02:37