Let U be the set of all polynomials of the form a0 a1 a2x2

Let U be the set of all polynomials of the form a_0 + a_1 + a_2x^2 + a_3x^3 for which a_0 = 0 and V be the set of all polynomials of the form a_0 + a_1x + a_2x^2 + a_3x^3 for which a_0, a_1, a_2, a_3 are integers. Let P_3 be set of all polynomials of degree

Solution

ANSWER: only U is subspace of P3

given P3 is a set of all polynomials of degree <= 3 with real coefficient

only U is subspace of P3

zero check
0 = 0 + 0x + 0x2 + 0x3 W since a0 = 0.
• closure under vector addition
Let p W = p(x) = a0 + a1x + a2x^2 + a3x^3, where a0 =0
= p(x) = a1x + a2x^2 + a3x^3.
Let q W = q(x) = b0 + b1x + b2x2 + b3x3, where b0 = 0
= p(x) = b1x+b2x^2 + b3x^3

Then
(p + q)(x) = p(x) + q(x)
= (a1+b1)x+(a2 + b2)x2 + (a3 + b3)x3.
Therefore, p + q U .

closure under scalar multiplication
Let c be a scalar and let p U = p(x) = a1x+a2x^2 + a3x^3. Then,
(cp)(x) = cp(x) = (ca1)x+(ca2)x^2 + (ca3)x^3.
Therefore, cp U .
Since U P3 contains the zero polynomial and is closed under vector addition and
scalar multiplication, U is a subspace of P3.

given v = a0 +a1x +a2x^2 +a3x^3

zero check:

0 =a0 +0x+0x^2+0x^3

0 =a0

so not true