Final answer:
The set of polynomials with the given condition on the coefficients satisfies all three subspace properties and is therefore a subspace of the space P of all polynomials.
Step-by-step explanation:
To determine if the set of all polynomials of the form ao + a1x + a2x^2 + a3x^3 with coefficients satisfying the condition a0 + a1 + a2 + a3 = 0 is a subspace of the space P of all polynomials, we must check if it satisfies the three subspace properties:
- Closure under addition
- Closure under scalar multiplication
- Contains the zero vector (the zero polynomial in this case)
First, take any two polynomials p(x) = a0 + a1x + a2x^2 + a3x^3 and q(x) = b0 + b1x + b2x^2 + b3x^3 in our set. If we add them together, we get (a0 + b0) + (a1 + b1)x + (a2 + b2)x^2 + (a3 + b3)x^3. Since a0 + a1 + a2 + a3 = 0 and b0 + b1 + b2 + b3 = 0, we have (a0 + b0) + (a1 + b1) + (a2 + b2) + (a3 + b3) = 0. The sum is another polynomial in our set, so closure under addition is satisfied.
Next, for scalar multiplication, take any scalar c and a polynomial p(x) = a0 + a1x + a2x^2 + a3x^3 in our set. Multiplying by c, we get cp(x) = ca0 + ca1x + ca2x^2 + ca3x^3. Since a0 + a1 + a2 + a3 = 0, after scalar multiplication we have ca0 + ca1 + ca2 + ca3 = c*(a0 + a1 + a2 + a3) = c*0 = 0. Therefore, closure under scalar multiplication is also satisfied.
Last is the zero vector condition. The zero polynomial 0 + 0x + 0x^2 + 0x^3 obviously has coefficients that sum to zero, so it belongs to our set.
Since our set meets all three conditions, it is indeed a subspace of the space P of all polynomials.