1. ## need help...about vector space and span

how do i solve this?

let V be P2(vector space of all polynomials of degree=2 including the zero polynomial)
S={P1(t), P2(t)}
P1=t^2+2t+1
P2=t^2+2
does S span P2 ?
THANKS!

2. Consider a general 2nd-degree polynomial $a_{1}t^2 + a_{2}t + a_{3}$. We must show that it can be written as a linear combination of the vectors in S:
$\begin{array}{rcl}a_{1}t^2 + a_{2}t + a_{3} & = & c_{1}P_{1}(t) + c_{2}P_{2}(t) \qquad \text{For some arbitrary scalar } c_{1}, c_{2} \\ & = & c_{1}(t^2 + 2t + 1) + c_{2}(t^2 + 2) \\ & \vdots & \\ & = & (c_{1}+c_{2})t^2 + (2c_{1})t + (c_{1} + 2c_{2}) \end{array}$

So you can conclude now?

3. Yes by definition?
S spans W = {aP1 + bP2} which includes element P2.

4. Originally Posted by debs19
how do i solve this?

let V be P2(vector space of all polynomials of degree=2 including the zero polynomial)
S={P1(t), P2(t)}
P1=t^2+2t+1
P2=t^2+2
does S span P2 ?
THANKS!
Originally Posted by o_O
Consider a general 2nd-degree polynomial $a_{1}t^2 + a_{2}t + a_{3}$. We must show that it can be written as a linear combination of the vectors in S:
$\begin{array}{rcl}a_{1}t^2 + a_{2}t + a_{3} & = & c_{1}P_{1}(t) + c_{2}P_{2}(t) \qquad \text{For some arbitrary scalar } c_{1}, c_{2} \\ & = & c_{1}(t^2 + 2t + 1) + c_{2}(t^2 + 2) \\ & \vdots & \\ & = & (c_{1}+c_{2})t^2 + (2c_{1})t + (c_{1} + 2c_{2}) \end{array}$

So you can conclude now?

what would be the exact answer?
do i need to use some variables like a b c and then solve for c1 and c2?
thanks

5. All you need for this question is the fact that V has dimension 3, and so cannot be spanned by a set with only 2 elements. I think the question has confused people because you have used P2 for both one of the elements of S and for the set of all polynomials with degree less than or equal to 2