Question

Determine which properties of a vector space are met by the set Q2 of polynomials with real coefficients that have degree equal to 2, together with the usual definition of addition and scalar multiplication for polynomials. (Select all that apply.) Property 1: If vand v2 are in V, then so is v. + V2. Property 2: If c is a real scalar and v is in V, then so is cv. Property 3: There exists a zero vector O in V such that 0 + v = v for all v in V. Property 4: Property 3 holds and for each v in v there exists an additive inverse vector -v in v such that v + (-v) = 0 for all v in V. Property 5(a): For all v4 and v2 in V, we have v1 + V2 = V2 + v1. Property 5(b): For all V1, V2, and V3 in V, we have (V1 + V2) + V3 = V1 + (V2 + V3). Property 5(c): For all v1 and v2 in V and real scalars c1, we have C1(V1 + V2) = C1V1 + C1V2. Property 5(d): For all vi in V and real scalars C1 and C2, we have (C1 + 2)v1 = C1V1 + C2V1. Property 5(e): For all vị in V and real scalars c1 and c2, we have (C102)V1 = C1(C2v1). Property 5(0): For all vy in V, we have 1.V. - V. none of these

Answer 1

The set Q2 of polynomials with real coefficients that have degree 2 satisfies all proposed properties of a vector space when considering the usual definitions of addition and scalar multiplication for polynomials, thus confirming it is indeed a vector space.

Step-by-step explanation:

We must determine which properties of a vector space are satisfied by Q2, the set of polynomials with real coefficients that have degree exactly 2, when considering the usual definitions of addition and scalar multiplication for polynomials. For vectors, addition is both associative and commutative, and multiplication by scalars is distributive.

Property 1 states that if v1 and v2 are elements of the vector space V, then so is their sum v1 + v2. This is true for Q2, as adding two quadratic polynomials results in another quadratic polynomial.

Property 2 indicates that for any real scalar c and vector v in V, the scaled vector cv is also in V. In Q2, multiplying a polynomial by a real scalar produces another polynomial of degree 2, so this property holds.

Property 3 involves the existence of a zero vector O in V such that O + v = v for any v in V. For Q2, the polynomial 0x^2 + 0x + 0 serves as the zero vector and satisfies this property.

Property 4 states that every vector v in V has an additive inverse -v such that v + (-v) = O. This is true for Q2 since the additive inverse of any polynomial is simply the polynomial with all coefficients negated.

Properties 5(a) through 5(f) relate to the algebra of vectors, outlining the laws of vector addition and scalar multiplication. All elements from Property 5(a) (commutative law of addition), Property 5(b) (associative law of addition), Property 5(c) (distributive law of scalar multiplication over vector addition), Property 5(d) (distributive law of vector addition over scalar addition), Property 5(e) (associativity of scalar multiplication), and Property 5(f) (scalar identity) are fulfilled by the set of quadratic polynomials, Q2, under usual addition and scalar multiplication.