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Given two sets:$S_1=\{(1,-2,1,0),(0,5,-2,1),(2,1,0,1)\},S_2=\{(3,-1,1,1),(1,3,-1,1)\}$. Check if $S_1$ and $S_2$ span the same subspace of the vector space $\mathbb R^4$.

Rank of a matrix formed of vectors in $S_1$ and $S_2$ is $2$.

Question: Does this mean that $S_1$ and $S_2$ span $\mathbb R^2$?

From the row echelon form of a matrix formed of vectors in $S_1$, we can see that $span(S_1)=span\{(1,-2,1,0),(0,5,-2,1)\}$ (third vector is dependent).

Now $$span(S_1)=\{c_1(1,-2,1,0)+c_2(0,5,-2,1):c_1,c_2\in\mathbb R\}=\{(c_1,-2c_1+5c_2,c_1-2c_2,c_2):c_1,c_2\in\mathbb R\}$$

From the row echelon form of a matrix formed of vectors in $S_2$, we can see that $$span(S_2)=\{c_3(3,-1,1,1)+c_4(1,3,-1,1):c_3,c_4\in\mathbb R\}=\{(3c_3+c_4,-c_3+3c_4,c_3-c_4,c_3+c_4):c_3,c_4\in\mathbb R\}$$

What confuse me is that from the rank of matrices formed from $S_1$ and $S_2$ we get that they span $\mathbb R^2$, but when forming linear combinations we see that they span different subspaces of $\mathbb R^4$.

So, finding the matrix ranks gives false answer?

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Your first question needs to avoid abuse notation as $\mathbb{R}^2 \not\subset \mathbb{R}^4$, rather those two vectors span a subspace isomorphic to $\mathbb{R}^2$. As for your other question, yes, $S_1$ and $S_2$ both span isomorphic spaces to $\mathbb{R}^2$ (or more precisely, subspaces of 2 linearly independent vectors in $\mathbb{R}^4$.), but they are NOT the same subspaces, and this is obviously allowed (why? can you think of an example?)

For example, perpendicular planes of $\mathbb{R}^2$ in $\mathbb{R}^4$ are clearly not the same, but require two vectors to span them.

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  • $\begingroup$ How to describe $span(S_1)=\{(c_1,-2c_1+5c_2,c_1-2c_2,c_2):c_1,c_2\in\mathbb R\}$? Is it $\mathbb R^2$ or $\mathbb R^4$? $\endgroup$ – user300048 Mar 27 '16 at 19:45
  • $\begingroup$ I think you should describe it as a subspace of $\mathbb{R}^4$, NOT $\mathbb{R}^2$ (to avoid confusion of notation). $\endgroup$ – q.Then Mar 27 '16 at 19:46
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Sure, $S_1$ and $S_2$ each span a copy of $\mathbb{R}^2$. But not the same copy : there are lots of $\mathbb{R}^2$ (ie planes) lying in $\mathbb{R}^4$.

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  • $\begingroup$ Are they spanning $\mathbb R^2$ or $\mathbb R^4$? $\endgroup$ – user300048 Mar 27 '16 at 19:41
  • $\begingroup$ It doesn't make sense to say that they "span $\mathbb{R}^2$", because there is so special copy of $\mathbb{R}^2$ inside $\mathbb{R}^4$. What you can say is that they each span a subspace isomorphic to $\mathbb{R}^2$. $\endgroup$ – Captain Lama Mar 27 '16 at 19:48
  • $\begingroup$ But then how to explain rank of matrices being $2$? $\endgroup$ – user300048 Mar 27 '16 at 19:50
  • $\begingroup$ By the fact that they span a subspace of dimension $2$. If the matrices had rank $3$, they would each span a subspace of dimension $3$, ie a copy of $\mathbb{R}^3$. $\endgroup$ – Captain Lama Mar 27 '16 at 19:52

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