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I am not entirely sure about this concept. Suppose that we have two sets of vectors say ${a,b,c}$ and ${u,v,w}$. The span of these two vector sets are equal to each other. If one set of vectors ${a,b,c}$ is linearly independent/or dependent will the other set of vectors {u,v,w} be linearly independent/or dependent i.e if one is linear independent will the other be linear dependent if we know their spans equal?

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closed as off-topic by Namaste, Simply Beautiful Art, José Carlos Santos, Isaac Browne, Adrian Keister Aug 3 '18 at 0:04

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If $\{a,b,c\}$ is a L.I. set, then $\text{dim}\,\text{span}\{a,b,c\} = 3$, and therefore since $\text{span}\{a,b,c\}=\text{span}\{u,v,w\}$, we must have that $\text{dim}\,\text{span}\{u,v,w\} = 3$.

Now since there are only 3 elements in $\{u,v,w\}$ and $\text{dim}\,\text{span}\{u,v,w\} = 3$, we must have that $\{u,v,w\}$ is a L.I. set also.

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Let the two "sets" of vectors you are talking about be denoted by $A$ and $B$. Indeed, if $S = span(A) = span(B)$, then you should be looking at the dimension of the $S$. Assuming $n \geq 3$ (sizes of the vectors), we have the following cases:

Cases:

If $\dim S = 1$: then both sets $A$ and $B$ are just linear combinations (scalar multiples) of one vector.

If $\dim S = 2$: then both sets $A$ and $B$ have two linearly independent vectors and the third is just a linear combination of two others.

If $\dim S = 3$: then both sets $A$ and $B$ have three linearly independent vectors and hence no vector could be written as a linear combination of the other.

$\dim S$ could not be greater than $3$ since we only have three vectors

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