Let p(x) = 2-x and q(x) = x. Using the inner product space = integral 1 to -1 pqdx find all polynomials r(x)=a+bx in P1(R)

Q: Use the inner product (p, q) = aobo + azb1 + azb2 to find (p, a), ||P||, |||, and d(p, q) for the…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: 1. Factorize each of the following polynomials over Z5 [x]. a) f(x) = x³ + 3x² + 3x + 2 b) f(x) = x¹…

A: The important thing to observe is that we have to check polynomials modulo 5 while factoring over…

Q: Prove that ithe polynomial PCO =-2xP +2x2+loxx t6 has and fachor azero at M= 3, and fachm s ala the…

A: The given polynomial is px=-2x3+2x2+10x+6. It is given that x=3 is a root of the polynomial. If x=3…

Q: Use the inner product = aobo+ aıb1+ azbz+ asb; to find , lIlp|], d(p, q), and then find the angle…

A:

Q: Use the inner product (p, q) = aobo + a₁b₁ + a₂b₂ to find (p, q), ||p||, ||9||, and d(p, q) for the…

A:

Q: The function f(z) is complex valued in the complex plane. f(z) is bounded and entire. Prove that…

A: As per the Louisville's theorem, every entire function which is bounded must be constant. Use the…

Q: 5A Given the inner product space of polynomials with real coefficients, degree ≤n (P₁(R),+,., ° ),…

A: The angle between the vectors in inner product space:

Q: Let A = 1 -X -X X 1 -x X X -1 Find x such that A is invertible.

A:

Q: Is the set of polynomials {-1+ 3t – 21², 4 – 10t + 312, 4 – 8t – 212} independent? Answer this…

A:

Q: Use the inner product (p, q) = abo + a,b, + a,b, to find (p, q), ||p ||, ||9||, and d(p, q) for the…

A:

Q: If {v, w} is independent, find conditions on the scalars k1 and k2 so that the set {k,v + w, v +…

A:

Q: Consider p(x) = 3x - 2 and g(x)=x+7. The evaluation inner product is defined as…

A:

Q: Let f(x)= x3 + x2 + x +1  ∈ Z2[x]. Write f(x) as a product of irreduciblepolynomials over Z2.

A: Given polynomial is: f(x)=x3+x2+x+1 Since, f(1)=0 So, x-1 is a factor. Now, factorize the…

Q: a) Find a Lagrange interpolating polynomial in the form of P, (x) = ax³ + bx² + cx+d befitting the…

A:

Q: Prove that (Z(sqrt(2)), +, ×) is an integral domain.

A: We know that , ℤ2 = a + b2 / a , b∈ℤ . Clearly , ℤ2 forms a ring. We know that , a commutative ring…

Q: Use the inner product (p, q) = aobo + a₁b₁ + a₂b₂ to find (p, q), ||pl|, |la||, and d(p, q) for the…

A:

Q: Let P₂ (R) be the space of all polynomials of order 2 and f: P₂ (R) → P₂(R) be the linear map…

A: We are given that f is a linear map.         f(1) = 1 + x        f(x) = 3 - x2         f(x2) = 4 +…

Q: b) Let P2(R) be the polynomial space of all polynomials of degree less than or equal to 2 whose…

A: To find out which of the following sets are linearly independent .

Q: Consider the linear operator T:R3→R3, given by T(v)=A⋅v, where [image]: What is the minimal…

A: Consider λ is the eigenvalue of the given matrix A. Therefore, 1-λ2301-λ100-1-λ=0. Expand the…

Q: Find a polynomial of degree three (that is, a polynomial of the form p(t) = a+btct²+dt³) whose graph…

A: Please see the attached images for the solution. Graph using desmos (for verification only)   If…

Q: = Let p(x) = x³ x² + 2x + 3, q(x) = 3x³ + x² − x − 1, r(x) = x³ + 2x + 2, and s(x) = 7x³ + ax² +5.…

A:

Q: Use the inner product (p, q) = abo + a,b, + a,b, to find (p, q), ||p ||, ||9||, and d(p, q) for the…

A: Since you have posted a question with multiple subparts, we will solve the first three subparts for…

Q: Use the inner product (p, q) = aobo + a¡b1 + azb2 to find (p, q), ||P||, ||g||, and d(p, q) for the…

A: Given polynomials are: p(x) = 3 - x + 2x^2 q(x) = x - x^2 We know that the inner product is given…

Q: factor x as a product of three linear polynomials in Z/30Z[x] by using Chinese remainder theorem

A: Ans- To factor x as a product of three linear polynomials in Z/30Z[x], we first need to factor 30…

Q: 2. (a) Define the dot product of two polynomials p(x) = ao+a₁x+a₂x²+a3x³ and q(x) = bo+b₁x+b₂x²+b3x³…

A:

Q: a) Factor a +x³ + x into irreducible polynomials in Z2[r];

A: i) Linear factors are irreducible. ii) Just substitute values in Zp and see the expression becomes…