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I need a little help with this question: Find a basis and dimension of the subspace R^n spanned by the following set:

{(1,3),(-1,2),(7,6)} (n=2)

I have tried attempting the question by putting the vectors in an augmented matrix and reducing it to row echelon form followed by taking the leading entries as the basis. I got {(1,3),(-1,2)} as my answer for the basis and 2 as the dimension. However, the solution provided {(1,0),(0,1)} as the answer for the basis.

I do not know where I'm going wrong with my answer. I would really appreciate it if I could get a little help on this question.

Thank You! :)

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  • $\begingroup$ Hint: is a basis unique? $\endgroup$ – IdentityMatrix Apr 21 '18 at 14:27
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Both answers are correct. The space spanned by those vectors is $\mathbb{R}^2$, and both $\bigl((1,0),(0,1)\bigr)$ and $\bigl((1,3),(-1,2)\bigr)$ are bases of $\mathbb{R}^2$.

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  • $\begingroup$ How did they manage to get {(1,0),(0,1)} as one of the solutions? $\endgroup$ – Jenny Apr 21 '18 at 14:28
  • $\begingroup$ @Jenny It's perhaps the most natural basis of $\mathbb{R}^2$. $\endgroup$ – José Carlos Santos Apr 21 '18 at 14:30
  • $\begingroup$ So am I right to say as long as I put the vectors in an augmented matrix, reduce it to row echelon form and take the columns of the leading entries as the basis, I should be fine? $\endgroup$ – Jenny Apr 21 '18 at 14:37
  • $\begingroup$ @Jenny Yes, if you also delete eventual null vectors. $\endgroup$ – José Carlos Santos Apr 21 '18 at 14:54
  • $\begingroup$ Thank you José for the explanation! :) $\endgroup$ – Jenny Apr 21 '18 at 14:58

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