a) Let a = V2+ v2 i) | Determine the minimal polynomial of a over Q. ii) | Write in terms of a Q-basis of Q(a) iii) Determine the minimal polynomial of a over Q(V

a) Find , \pl, d (p, q), and then find the angle between them for the polynomials in P2 if p(x) = -3x² + 2x – 5 , and q(x) = -x+5x? b) Verify that the basis set S = {(3,0,4), (-4,0,3), (0,1,0)} is orthogonal then find the coordinates of the vector u = (-5,5,1) using inner product

Let P² be the vector space of all polynomials with real coefficients of degree at most 2, with the usual operations. a) Show that the set of vectors S = {x² + 1, 3x − 14, -5x²9x+37} is linearly dependent. b) Find a basis for P2 which includes the first two vectors of S (that is, v₁ = ²+1 and v₂ = 3x - 14).

Show that the set of all real polynomials with a degree n < 3 associated with the addition of polynomials and the multiplication of polynomials by a scalar form a vector space.

Using linear algebra, show that G={p1(x)=1−x+x^2, p2(x)=x, p3(x)=x+x^2} is a basis of P2, the vector space of polynomials of degree at most two with real coefficients.

Show that S = {t+1,t-1,(t-1)2} be a basis of P2, the vector space of all second-degree real polynomials. Express the polynomial pt=2t2-5t+9 as a linear combination of elements of S.

Let P 2 be the vector space of all polynomials of degree less than or equal to 2, under the usual vector addition and scalar multiplication. Let S = {f (x) ∈ P 2 | if deg(f ) = 2, then f has two complex roots}.  (a) Is "0 ∈ S. Why or why not? (b) Is S closed under addition? If S is closed under addition, prove it. If S is not closed under addition, provide a counterexample.   (c) Is S closed under scalar multiplication? If S is closed under scalar multiplication, prove it. If S is not closed under scalar multiplication, provide a counterexample.

Which of the following sets of polynomials form a basis for P2? (i) 1, p2 1- 5х, рз = x, P4 (i1) рі 1+ 2x + 5x-, p2 4 +x 1 + 9x + x2 || (iii) pi = 1+x, p2 = 9x + x², p3 A) (i) and (ii) only (B) (i) only (C) (ii) only (D) (ii) and (iii) only (E) (i) and (iii) only (F) (iii) only G) none of them (H) all of them

Find a basis for the vector space of all quadratic polynomials p(x) = ax²+bx+c, that satisfy p(−1) = 0.

let p(x) = a0+a1x+a2x2 and  q(x) = b0+b1x+b2x2 be vectors in P2 with inner product <p,q> = a0b0+a1b1+a2b2 Consider the polynomials in the set {−1+x2,2+x+x2} 1)Show that the polynomials in the set do not form an orthonormal set. 2)Use the Gram-Schmidt orthonormalization process to form an orthonormal set from these polynomials. Keep the order, and show work for the orthogonalization, then normalization. 3)Do the polynomials in the orthonormal set (from question 2) form a basis for P2 ? Why or why not?   i mainly need the answer for 3 but if u can please answer the other 2. also please make it clear what part ur answering