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So we have set of two following vectors $x_1=(2,0,-1,0) $ and $ x_2=(4,1,-2,1)$ Now we need to extend this set of vectors to the basis of vector space $\mathbb{R}^4$. There are few ways to solve this, and i've solved it by using vectors of the canonical base of $\mathbb{R}^4$ and i added them to this set and then i formed system of linear equations in order to determine which four of them are linearly independent (there are several possible combinations, however, in this case it way sufficient to find only one of them) and my solution was $ x_1, x_2, e_1, e_4$ however, i am interested in different approach to this problem (the one i found but i don't understand completely) which says that i should represent given vectors as a linear combination of vectors from the canonical base of $\mathbb{R}^4$. So actually we have $x_1=2e_1-e_3$ and $x_2=4e_1+e_2-2e_3+e_4$ and then, next thing i have is $e_3=2e_1-x_1$ from the first equation and $e_4=x_2-2x_1-e_2$ from the second equation, now, for some reason, this above proves that $x_1,x_2,e_1,e_2$ is basis of $\mathbb{R}^4$ and it is, just like $x_1,x_2,e_1,e_4$ is too. But i'd like to understand why expressing $e_3$ as a linear combination of $e_1$ and $x_1$ and $e_4$ as a linear combination of $x_1$, $x_2$ and $e_2$ proves that $x_1,x_2,e_1,e_2$ is the basis of vector space $\mathbb{R}^4$?

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You showed $e_3$ and $e_4$ can be expressed in terms of $x_1,x_2,e_1,e_2$. This means all the vectors in the set $B=\{e_1,e_2,e_3,e_4\}$ are all contained in the span of the set $A=\{x_1,x_2,e_1,e_2\}$, and so this set spans $\Bbb R^4$ since $B$ spans $\Bbb R^4$. You now have $4$ vectors which span a $4$-dimensional space, and so they are Linearly independent, and so $A$ forms a basis of $\Bbb R^4$.

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  • $\begingroup$ let me see if i understood you well, if i have a set of 4 vectors from $\mathbb{R}^4$ and if i can express every vector of canonical basis of $\mathbb{R}^4$ as a linear combination of those 4 vectors that means that those four vectors are basis of $\mathbb{R}^4$, is that right? $\endgroup$ – cdummie Apr 22 '17 at 14:41
  • $\begingroup$ Yes, that's correct. $\endgroup$ – John Doe Apr 22 '17 at 15:18

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