Suppose that p0, p1, . . . , pn are polynomials in Pn(R), where Pn is the set of all polynomials of degree less than or equal to n, such that pj(2) = 0 for all j. Prove that {p0,p1,...,pn} is not linearly independent in Pn(R)

Q: Prove that there is only one polynomial P3(x) among all polynomials of degree < 3 that satisfy the…

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Q: z -7 2. prove that: Im(z)= 2i

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Q: a b Let A=| c d Then the characteristic polynomial P() of A is given by P(N=x2-Trace(A)A – det(A).…

A: Trace of the matrix is equal to the sum of its diagonal elements.  The formula to find…

Q: Exercise 47 (a) Show that there exists a unique polynomial po of degree < 2 such that po(-1) = 1,…

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Q: Consider the polynomials p, (1) = 6 +r and p () =6-?. Is (p,. P2} a linearly independent set in P,?…

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Q: a) Show that the cubic polynomials P(x) = 3 − 2(x + 1) + 0(x + 1)(x) + (x + 1)(x)(x − 1) and Q(x) =…

A: Given two cubic polynomial,

Q: 20 Lagrange's polynomial p1 (x) that interpolates f(x) = at a = 2 and a = 5 is Select one: O a.pr(z)…

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Q: a. Let V be the the set of polynomials of degree less than or equal to 2, with operations defined as…

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Q: (a) Let p(x) be a polynomial with coefficients in R, and let M abe the 3 a 1 0 0 a 1 0 0 a Show that…

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Q: For which real values of k do the following polynomials form a linearly dependent set in P,. 2-2x+kx…

A: Explained below...

Q: Using MOSA find the polynomials that will approximate the solution to: y' =x + y, y(1)-1. y1(x) C a)…

A: The given differential equation is y'=x+y with y1=1. Here x0=1, we have y=1. Therefore y0x=1.…

Q: (a) Let p(x) be a polynomial with coefficients in R, and let M abe the 3 x 3 matrix а 1 Ma := a 1 а…

A: Given that p(x) be a polynomial with coefficients in R and Ma be the 3×3 matrix Ma=a100a100a

Q: Let Po. P1, and p2 be the orthogonal polynomials described below, where the inner product on P, is…

A: Given: p0(t)=1p1(t)=3tp2(t)=t2-2The points are: -2, -1, 0, 1, 2Let p(t)=2t3Then at…

Q: Find det (A) given that A has p(1) as its characteristic polynomial. p(1) = 13 – 21² + 1 + 29 det…

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Q: [5 2 1] 1 1 2 7 15 2 What is the coefficient of X in the characteristic polynomial of the matrix A =…

A: Given matrix A=5211127152

Q: Consider the structure R = (IR, +, x, <). We say that a set C CR is definable in R if there is a…

A: Given: R=ℝ,+,×,&lt; C⊆ℝ is definable in R if there is a formula φx such that c∈C if and only if φc…

Q: Which of the following statements hold for the minimal polynomial for a matrix A? (i) The minimal…

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Q: 1 2 3 0 4 5 0 0 6 6. (a) Let A = Find the characteristic polynomial of A and use it (and…

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Q: Prove that the dimension of Pn(R), the set of all polynomials of degree less than or equal to n, is…

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Q: 2. Let p1(t) = 1, p2(t) = 2t, p3(t) = -2+ 4t2 and p4(t) = -12t + 8t3 be polynomials is P3 (these are…

A: (1)Q-2       Given that p1(t) = 1  , p2(t) = 2t       , p3(t) = -2+4t2p4(t) = -12t+8t3Result:Let dim…

Q: 8. (Liouville's theorem, Cauchy estimates and the maximum modulus principle) (a) Suppose that f is…

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Q: Find det (A) given that A has p(A) as its characteristic polynomial. P(A) = 14 – 13 + 20 det (A) =…

A: Given that, ρλ=λ4-λ3+20 And, detλI-A=λn+c1λn-1+....+cn Where -1ndetA=cn

Q: A X 5. Let G = be a block matrix where A and B are both 2 x 2 blocks. Show that В det G = (det…

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Q: Matrix A has characteristic polynomial p(2) = (-45-2)(5-2)(18-2). -7 6 28 A = 6 2 -14 28 -14 -17…

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Q: A, characteristic polynomial A4 (x) = (x – 2) (x + 2)°, Let be a rational matrix whose minimal…

A: We will find the Jordan canonical form or Jordan normal form for a matrix A whole characteristic and…

Q: 10) Find the characteristic polynomial and the minimal polynomial 4 0 0 0 4 0 0 -4 0 0 0 -4

A: Given the matrix, A=4000040000-40000-4 To find the characteristic polynomial, consider A-tI, where A…

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: Find det (A) given that A has p(A) as its characteristic polynomial. P(A) = A4 – A3 + 11 det (A) =…

A: Here given the characteristic equation of matrix A is  pλ=λ4-λ3+11 Since the order of characteristic…

Q: [1 a b Let A = 0 5 2 0 0 1 (а) (i) Determine the condition of a and b so that A is diagonalisable.…

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Q: Using the Rodriguez formula obtain the Legendre polynomials Pn(x) for n=0,1,2,3,4,5

A: We have to find the Legendre polynomial Pnx by using  Rodriguez formula for n=0, 1, 2, 3, 4, 5. The…

Q: (a) Demonstrate that a polynomial is completely determined by its coefficients. That is, if n and m…

A: (d) Given that m1,m2,b1,b2 are real no. we need to express lm1,b1.lm2,b2 in quadratic form. We'll…

Q: (b) Find an expression az + ba? + ca +d so that this new polynomial is orthogonal to all the…

A: Solution of part (b): The inner product on continuous functions on -2,2 is defined by…

Q: Let P(z) = z" +am-12"-1 + ... + a1z + ao be a polynomial of degree n 2 1 with a; # 0 for at %3D…

A: We will solve Q9 as mentioned. Given Pz=zn+an-1zn-1+…+a1z+a0 be a polynomial of degree n≥1 with ai≠0…

Q: 8. (Liouville's theorem, Cauchy estimates and the maximum modulus principle) (a) Suppose that f is…

A: (a) Given that, fz is an entire function Thus, we can write the fz as a power series around zero,…

Q: [0, 2] aralığı üzerinde f(x) = Consider the function f(x) = 0-² + O²+1 0 - +1 O r O 2+1 fonksiyonunu…

A: Given function is fx=x21+x2, x∈0,2. We have to construct the Lagrange's interpolation polynomial…

Q: Let Po. P4, and p, be the orthogonal polynomials described below, where the inner product on P, is…

A: Let po. P₁, and p, be the orthogonal polynomials described below, where the inner product on P, is…

Q: Find all values of z such that z5 =1 - i.

A: Given: z5=1-i, To find: All values of z.

Q: Let -3 1 0 -1 A = -3 -4 4 -3 -4 4 Find the dimension of the kernel and the dimension of the image of…

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Q: (a) Describe the kernel and the image of Jn. (b) Use this to show that J, is diagonalisable when n #…

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Q: 163 290 -68 2. Let B =-150 -262 60 -245 -414 91 This matrix has characteristic polynomial - (A- 6)…

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Q: 5. Let f be the polynomial f(z) = z° + z5 – 8z3 + 2z + 1. Find the number of zeros of f in the…

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Q: 5. Let W be the space of polynomials p(x) with real coefficients of degree < 10, satisfying p(0)…

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Q: 2. (a) Show that the polynomials p, (x)=1+x, p,(x)=1+2x–x², p;(x)=x+x² are Р. linearly independent.…

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Q: 5. Find the Newton's interpolating polynomial passing through the poin (x1. yı) = (-4, 5), (x2, y2)…

A: Given that: Three points (-4,5), (-2, 3), (0, 9)

Q: 3. Let T,(x)= f(a) + f '(a)(x-a)+ "(a 2(x-a)² +² a)(x-a)³ be the Taylor polynomial of degree three…

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Q: 3 1 1 -1 Let A 2 4 -2 -1 -1 3 Find the characteristic and minimal polynomials of A.

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Q: Which of the following statements hold for the minimal polynomial for a matrix A? (i) The minimal…

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Q: Lagrange's polynomial p1 (1) that interpolates f(x)= at z = 2 and z = 5 is Select one: O a. pi(2) 3…

A: Here there are two points for interpolation, so we will get a one degree interpolating polynomial.

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.