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But do appear to be linear combinations? $u$ and $v$ are $3$-component vectors. The question posed is:

Find two vectors in $\operatorname{span}\{u,v\}$ that are not multiples of $u$ or $v$ and show the weights on $u$ and $v$ used to generate them.

I.e. I am looking for two vectors $au + bv$ and $cu + dv$, and I take it from the wording of the question that the one restriction on these coefficients is that they are all non-zero (or I'd just have a multiple of $u$ or $v$).

Some things I understand or have already demonstrated to myself:

  • $u$ and $v$ are linearly independent
  • together they describe a plane in $3$-dimensional space
  • the problem setter wants vectors that are 'off' the directions described by $u$ and $v$ (i.e not parallel).

From this, I thought any two linear combinations would be fine, but none of the ones I've submitted are acceptable. Is there something in this question that I have misunderstood? I would really appreciate any help understanding what is being asked (rather than being given the coefficients).

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    if $u,v$ are linearly independent, $u+v,u-v$ always works. perhaps the software will only recognize this as the "correct" answer (even though there are many others).2012-03-06

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