Question

given P2 as the vector space of all real polynomials of degree less than or equal to two. Let W be a subspace of P2 specified by W= {a0 + a1x + a2x^2 where a0 -a1 =0}. Determine whether W is a subspace of P2.

Answer 1

Let A, B ∈ P2 such that

$A = a_0 + a_1x + a_2x²$ , where $a_0 = a_1$

$B = b_0 + b_1x + b_2x²$ , where $b_0 = b_1$

Let α be any scalar from the field

$A+αB = a_0 + a_1x + a_2x² + α(b_0 + b_1x + b_2x²)$

$A+αB = a_0+αb_0 + a_1x+αb_1x + a_2x²+αb_2x²$

$A+αB = (a_0+αb_0) + (a_1+αb_1)x + (a_2+αb_2)x²$

Since $b_0 = b_1 => αb_0=αb_1$

Also, we know that $a_0 = a_1$

$=> a_0+αb_0=a_1+αb_1$

Thus, $A+αB = (a_0+αb_0) + (a_1+αb_1)x + (a_2+αb_2)x²$ , where $a_0+αb_0=a_1+αb_01$

Hence W is a Subspace of P2