# Thread: Does S span P_2?

1. ## Does S span P_2?

Consider the set {1+x^2, 4-2x, 1+x+x^2} of three vectors in the Vector Space P_2. Does S span P_2?

I proved that it was closed under vector addition and multiplication is that enough or is there more?

2. Originally Posted by matt15math
Consider the set {1+x^2, 4-2x, 1+x+x^2} of three vectors in the Vector Space P_2. Does S span P_2?

I proved that it was closed under vector addition and multiplication is that enough or is there more?
The idea here is to test and see if these functions are linearly independent. If they are linearly independent, they also span the set.

If we put the coefficients of the polynomials in a matrix and take its determinant, we see that

$\begin{vmatrix}1 & 0 & 1\\ 4 & -2 & 0\\1 & 1 & 1\end{vmatrix}=\begin{vmatrix}-2 & 0\\ 1 & 1\end{vmatrix}+\begin{vmatrix}4 & -2\\1 & 1\end{vmatrix}=-2+6=4\neq 0$.

Therefore, the polynomials in set $S$ are linearly independent and span $P_2$.

3. Originally Posted by matt15math
Consider the set {1+x^2, 4-2x, 1+x+x^2} of three vectors in the Vector Space P_2. Does S span P_2?

I proved that it was closed under vector addition and multiplication is that enough or is there more?

I suppose you meant to ask whether $Span(S)=P_2$, and since $\dim P_2=3$ , three vectors span it iff they're a basis iff they're linearly independent, so you've to check whether $a_1(1+x^2)+a_2(4-2x)+a_3(1+x+x^2)=0 \Longleftrightarrow a_1=a_2=a_3=0$

Tonio