What are the rules to determine if a set is a basis for V? So I have here an example, we let $S = \{t^2 + 1, t - 1, 2t + 2\}$
How can I determine if $S$ is a basis for $V = P_2$?
Also, do I need to prove that $S$ is linearly independent? I'm almost there, just need someone to actually point me in the right direction. Please help. 
 A: Usually, we need to prove two statements:


*

*$S=\{s_1,s_2,s_3\}$ is linearly independent, so any element in $S$ is not a linear combination of anothers, i.e.,
$$as_1+bs_2+cs_3=0$$
for scalars $a,b,c$ not all zero; and

*$V$ es generated by $S$ (or $S$ spans $V$); briefly $V=\text{span}(S)$. To do that, we pick an arbitrary element $v$ in $V$, and show that 
$$as_1+bs_2+cs_3=v$$
for scalars $a,b,c$ (possibly all zero).
In your case


*

*In order to prove that $S = \{t^2+1,t-1,2t+2\}$ is linearly independent, we need to show that the only solution for
$$a(t^2+1)+b(t-1)+c(2t+2)=0$$
is the trivial solution (i.e., $a=b=c=0$).
You can "translate" to linear system equations. Note that
$$xt^2+yt+z\sim(x,y,z)$$ and so
$$\begin{matrix}t^2&&+1&\sim&(1, \ \ \ 0,\ \ \ 1)\\ &t&-1&\sim&(0, \ \ \ 1,-1)\\ &2t&+2&\sim&(0, \ \ \ 2, \ \ \ 2)\end{matrix}$$
Thus, we need to solve
$$a(1,0,1)+b(0,1,-1)+c(0,2,2)=0$$
i.e.,
$$\begin{matrix}a&\cdot&\color{red}{1}&+&b&\cdot&\color{blue}{\ \ \ 0}&+&c&\cdot&\color{green}{0}&=&0\\ a&\cdot&\color{red}{0}&+&b&\cdot&\color{blue}{\ \ \ 1}&+&c&\cdot&\color{green}{2}&=&0\\ a&\cdot&\color{red}{1}&+&b&\cdot&\color{blue}{-1}&+&c&\cdot&\color{green}{2}&=&0\end{matrix}$$

*In the same fashion, we need to solve
$$a(1,0,1)+b(0,1,-1)+c(0,2,2)=(x,y,z)$$ for arbitrary $xt^2+yt+z$ in $P_2$, 
i.e.,
$$\begin{matrix}a&\cdot&\color{red}{1}&+&b&\cdot&\color{blue}{\ \ \ 0}&+&c&\cdot&\color{green}{0}&=&x\\ a&\cdot&\color{red}{0}&+&b&\cdot&\color{blue}{\ \ \ 1}&+&c&\cdot&\color{green}{2}&=&y\\ a&\cdot&\color{red}{1}&+&b&\cdot&\color{blue}{-1}&+&c&\cdot&\color{green}{2}&=&z\end{matrix}$$
If we can solve $a,b,c$ regard to $x,y,z$, then we are done. $\Box$
A: I think I got the answer. Correct me if I am wrong:
A set of vector S in a vector space V is called a basis of V if S spans V and S is linearly independent.
We just need to answer the question: Does S spans V? and Is S Linearly Independent?
A: Hint:  The determinat of the matrix $\begin{pmatrix}  1&0&1 \\ -1 &1&0\\2&2&0 \end{pmatrix}$ is not zero.
A: Hint: To prove linear independence you can use the Wronskian.  
