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If you have a set of vectors, V = {v1, v2, v3}, with each vector containing 3 elements (x,y,z) and you want to know if all of V spans a vector space, Rn, my understanding is that you want to set up a system of linear equations and set them = 0.

Is the reason for this because you are checking to see if the zero vector (0,0,0), the smallest subspace of a vector space, is a linear combination of V?

My thoughts come from the below:

If you want to check if one of the vectors, v1, is a linear combination of the other vectors, you would set up a system of linear equations where

v1 = x1v2 + x2v3

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Recall that by definition three vectors are linearly dependent if and only if

$$x_1v_1+x_2v_2+x_3v_3=0$$

with $x_i$ not all equal to zero.

If we consider only the system

$$v_1=x_1v_2+x_2v_3$$

we can only check that $v_1$ is a linear combination of $v_2$ and $v_3$ but we can't check if $v_2$ and $v_3$ are linearly dependent.

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  • $\begingroup$ Trivial example: $v_2=v_3=0$, $v_1\ne0$. For a not so trivial example, take $v_2=kv_3$ and $v_1$ not a scalar multiple of either. $\endgroup$ – amd Nov 2 '18 at 0:21
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Once you have three vectors you have a vector space which is the span of your vectors. It is the question of dimension of that vector space that brings the linear dependence or independent to the scean.

Your span is one, two ,three dimensional space , depending on how many of them are linearly independent assuming that they are not all zero vectors.

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