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For example the vector space $\Bbb R^3$.

I think that's a pretty good starting point. If I had three vectors, how would I go on acquiring the values that doesn't exist for $\Bbb R^3$?

Help would be appreciated.

F

FringleWins
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    Welcome to MSE. What do you mean by "the values that [don't] exist for $\mathbb{R}^3$? Also in the title, what are "values that [don't] span a vector space"? I can't understand this question at all. – saulspatz Jun 05 '20 at 18:01
  • if you have three vectors but they lie in the same plane, they do not span all of $R^3.$ In turn, this is accomplished by having a nontrivial linear relation among the three: vectors $u,v,w$ with real coefficients $a,b,c$ that are not all zero, with relation $au + bv + cw = 0$ as vectors. – Will Jagy Jun 05 '20 at 18:02
  • One quick way is to take some $u,v$ and then construct $w=u+v$ – Will Jagy Jun 05 '20 at 18:03
  • Hey, for example if I have the vectors $(1, -α, β), (α, -1, -α), (0, α - 1, β)$.

    I would like to find values for $α$ and $β$ such that they do not span $\Bbb R^3$.

     – FringleWins Jun 05 '20 at 18:16
  • You can put the vectors as the columns of a matrix and reduce. The three vectors span all of $\mathbb{R}^3$ if and only if the reduced matrix has 3 pivot columns if and only if the reduced row echelon form is the identity matrix. – twosigma Jun 05 '20 at 18:41
  • I am sorry, I appreciate the comment, but I don't quite understand. Like calculus, I am also trying to get back into linear algebra. It seems I have forgotten some concepts. I would appreciate if you could elaborate more, for my sake, if you will. Thanks. – FringleWins Jun 05 '20 at 18:48
  • You can see a post here about the column space and how it relates to matrix multiplication. To say the 3 column vectors span $\mathbb{R}^3$ means the column space of the matrix $A$ is $\mathbb{R}^3$. Since row operations preserve linear dependence among columns, row reduction is justified if you just want to find the dimension of the span of the columns (i.e. rank). – twosigma Jun 05 '20 at 19:30

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