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If I have a basis $(3,1,0,0,0),(0,0,7,1,0),(0,0,0,0,1)$ for a subspace of $\mathbb{R}^{5}$ and want to extend this collection of vectors to a basis for $\mathbb{R}^{5}$ is there any methodical, or algorithmic way to do this besides logically picking two linearly independent vectors to extend the set to a basis?

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    Are you familiar with Gram Schmidt orthogonalisation? – copper.hat Dec 21 '19 at 20:16
  • yea I'll have to review it, but I remember using the method we can find linearly independent vectors algorithmically –  Dec 21 '19 at 20:18
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    One (slow) way is to start with a known basis, such as that provided by the unit vectors $e_1,...e_5$. Then see if each is contained in the the span of the above list ($e_5$ is, trivially). If so, ignore, otherwise add to the collection. Rinse & repeat until finished. – copper.hat Dec 21 '19 at 20:19
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    @copper.hat You can test for all of them at once using Gaussian elimination, see my answer below. – xxxxxxxxx Dec 21 '19 at 20:25

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The standard way is to form a matrix with these vectors as columns, which has column space equal to $\mathbb{R}^{5}$: $$\begin{bmatrix} 3&0&0&1&0&0&0&0 \\ 1&0&0&0&1&0&0&0 \\ 0&7&0&0&0&1&0&0 \\ 0&1&0&0&0&0&1&0 \\ 0&0&1&0&0&0&0&1 \end{bmatrix}$$ Now you want to follow the method to get a basis for the column space: Figure out which are the pivot columns, and use those as a basis for $\mathbb{R}^{5}$ (note that with this setup, since your original three vectors were linearly independent, they will be pivot columns and so used as the first three vectors of the basis).

xxxxxxxxx
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