[Math] how to prove a spanning set of polynomial

linear algebravector-spaces

I am struggling so much understanding this concept of subspace and span.

The question is, Given that

$P2:W=\{(x+1)(ax+b)| a,b \in R\}$
show that
$\{x^2+x, x^2+2x+1\}$
is a spanning set of $W$.

I don't know if I got this concept right, but I've tried to do things by letting
$p(x)=x^2+x$
$q(x)=x^2+2x+1$
then multiplying them a coefficient $\alpha$ and $\beta$ each and adding to fit in with $W$.

but then I got an answer saying $a=\alpha + \beta$, $a+b = \alpha+ 2\beta, b=\beta$, which means… no solution? I am guessing? so this does not span $W$. Am I right?

Best Answer

$(x+1)(ax+b)=ax^2+(b+a)x+b.$

Claim: There exists $\alpha, \beta\in \Bbb R $ such that $$ax^2+(b+a)x+b=\alpha(x^2+x)+\beta( x^2+2x+1)\tag1$$ $$ax^2+(b+a)x+b=(\alpha+\beta)x^2+(\alpha+2\beta)x+\beta$$ Comparing coefficients, we get $$a=\alpha+\beta\tag2$$ $$b+a=\alpha+2\beta\tag3$$ $$b=\beta \tag4$$ Substituting $(4) $ in $(1) $ gives $\alpha=a-b$ (you can verify the solution using $(3)$). In other words, $\forall a,b\in\Bbb R $, there exists $\alpha=a-b, \beta =b $ such that $(1) $ holds. Hence $\{x^2+x,x^2+2x+1\} $ is indeed a spanning set.