How do I determine whether a set spans or does not span a vector space? $\{1+x^2,1+x+2x^2,x+x^2\}$ in $P_2$
Does the set span $P_2$?
I understand that the set is linearly dependent but how can I prove that the two independent elements cannot form every element in $P_2$? Does the set have to be linearly independent for it to span a particular vector space?
 A: A set can span a vector space without being linearly independent. However, if you know about dimensions then a set spanning the three-dimensional space $P_2$ must contain three linearly independent elements (and possibly additional elements).
Here, to show that the set is not spanning, you could also exhibit a single element of $P_2$ that is not writeable as linear combination of the given vectors. For example, can you see why $1$ is not in the span?
A: We Have a result, That 

"If a set is Linearly independent and has dimension same as that of Vector Space $V$, Then The set is a Basis for $V$." 

So I assume that your vectors are not Linearly Independent.
$1+x^2 \quad \qquad= 1(1)+0(x)+1(x^2)$
$1+x+2x^2 \quad= 1(1)+1(x)+2(x^2)$
$x+x^2 \quad \qquad= 0(1)+1(x)+1(x^2)$.
So the matrix of the coefficients

\begin{pmatrix}
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 2 & 1
\end{pmatrix}

Row reduce it to echelon form, You will get 2 non zer0 rows only, hence they are not linearly independent vectors. they dont span the vector Space.
A: It doesn't have to be linearly independent for a set to span a vector space. What you'll want to do is check if you can get any element from $P_2$ from a linear of combination of the elements in your set. In this case, $\{1+x^2,1+x+2x^2,x+x^2\}$.
