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If I have two 3x1 column vectors in a vector space V that are linearly independent, how can I make a system of 3 eqns whose solution will span V?

For example,

column vector [1,3,0], column vector [1,0,-1], so these vectors form V. What third vector can I make so that it will span S?

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Hint: find a vector orthogonal to both (e.g. take vector product $[1,3,0]\times [1,0,-1]$...)

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  • $\begingroup$ what?? im lost... could you just show me how to do it? $\endgroup$ – Math Dude Nov 16 '14 at 18:43
  • $\begingroup$ The vector product $[1,3,0]\times [1,0,-1]=[-3,1,-3]$ is orthogonal to both $[1,3,0]$ and $[1,0,-1]$. The triple $[1,3,0], [1,0,-1], [-3,1,-3]$ is a base for the whole space $\mathbb{R}^3$. For how to compute the cross product, see en.wikipedia.org/wiki/Cross_product $\endgroup$ – Milly Nov 16 '14 at 20:27
  • $\begingroup$ But how does that satisfy the solution space if it's orthogonal...? $\endgroup$ – Math Dude Nov 16 '14 at 20:41
  • $\begingroup$ they span the whole space. what do you mean by "satisfy the solution space"? If you have nontrivial equations, then the solution space is less than the whole space... $\endgroup$ – Milly Nov 17 '14 at 2:25

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