Very recently I started studying proof based linear algebra on my own and I am having a difficulty because there is no one to tell me whether if my proof is right or wrong. Please keep in mind that this is my first time dealing with proofs. The question is the following:
Let a system of vectors $v_1, v_2 ... v_r$ be linearly independent but not generating. Show that it is possible to find a vector $v_{r+1}$ such that the system $v_1, v_2, \dots, v_r, v_{r+1}$ is linearly independent.
Hint: Take for $v_{r+1}$ any vector that cannot be represented as a linear combination of $v_1,v_2,\dots, v_r$ and show that the system $v_1, v_2, \dots, v_r, v_{r+1}$ is linearly independent.
My attempt: question says $v_1,\dots,v_r$ is linearly independent so it has trivial coordinates $a_k$. $a_1v_1 + a_2v_2 + \dots + a_rv_r = 0; a_1=a_2=\dots=a_r=0$
I add $v_{r+1}$ to the set $\{v_1,v_2,\dots,v_r\}$ and see if they can form linear independence.
$a_1v_1 + a_2v_2 + \dots + a_rv_r + a_{r+1} v_{r+1} = 0$ but $a_1=a_2=\dots=a_r=0$ so $a_{r+1} v_{r+1} = 0$ is true if $a_{r+1} = 0$ (which makes it linear independent) as long as $v{r+1}$ is not the $0$ vector.
Therefore, there is a $v_{r+1}$ that makes a system $\{v_1,v_2,\dots,v_{r+1}\}$ linearly independent. My proof ends here.
**Extra question: In the beginning, it says system of vectors given are linearly independent but not generating. Those this mean it has basis but not all of the basis? Like if the space was R^3, then only 2 or 1 out of 3 basis are in the system?