linear algebra: span

Find a basis for the subspace V of P2, consisting of all vectors of the form at^2+bt+c, where c=a-b.

In the solution, it states that every vector in V is of the form at^2+bt+a-b, which can be written as a(t^2+1)+b(t-1). And that therefore the vectors t^2 +1 and t-1 span V.

I don't get why a(t^2+1)+b(t-1) means t^2 +1 and t-1 span V.. can someone please explain this to me?

Question

linear algebra: span

Find a basis for the subspace V of P2, consisting of all vectors of the form at^2+bt+c, where c=a-b.

In the solution, it states that every vector in V is of the form at^2+bt+a-b, which can be written as a(t^2+1)+b(t-1). And that therefore the vectors t^2 +1 and t-1 span V. 

I don't get why a(t^2+1)+b(t-1) means t^2 +1 and t-1 span V.. can someone please explain this to me?

anonymous · Answer

For vectors to span a subspace it means that you can create every vector in the subspace by a linear combination of the vectors. $a(t^2+1)+b(t-1)$ is a linear combination of those two vectors, so they span the subspace.

he66666 · Answer

Oh I see thanks! :)