1. Let P3 be the space of polynomials of degree at most 3, i.e., P3 = {po+ p1x + p2x² + p3x% : po, P1, P2, P3 € R}. Let T : P3 → P3 be the mapping defined by Tf (x) = f"(x) – 4f'(x) +f(x). Show A = {1, 1+x, (1+x)², (1+x)³} is a basis for P3. Find the matrix representation of T with respect to the ordered basis A = (1, 1+x, (1+x)², (1+x)³). %3D %3D

Q: Suppose f and g are non-constant, differentiable, realvalued functions defined on (−∞, ∞).…

A: We plug x=0 and y=0

Q: 6. Determine whether the following set of polynomials is linearly independent or not. (1 – t)?, t –…

A: Solution:-

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A: Irreducible polynomial and it's order

Q: f: X→ Y and g: Y→V be bijective functions. Show the following: ) gof: X→V is injective. ) gof: X→ V…

A:

Q: f: X→Y and g: Y→V be bijective functions. Show the following: ) gof: X→ V is injective. ) gof: X → V…

A:

Q: Show that f(x)=x\power{2}-3 is the minimal polynomial of \sqrt{3} over Q,

A: Let            We  have to  find the  minimal  polynomial of  the    over Let Then,  Suppose  that

Q: Prove that f(x) = 5x2 + 3x – 26 is θ(x2)

A: Given: f(x) = 5x2 + 3x – 26 is θ(x2)

Q: Given points pı = , and p3 in R², P2 let S = conv{pj,P2»P3}. For each linear functional f, find the…

A:

Q: The function

A: Given, the function f : ℂ→ℂ defined by                   f(z)=ez+e-zhas

Q: Suppose UCC is a domain, f, g = O(U). If |ƒ(z)|2 + |g(z)|² = 1 for all ZEU, show that ƒ and g are…

A: The solution to the problem is in the segment below.Explanation:To show that f(z) and g(z) are…

Q: 1. The map T : Mat2,2(R) → Mat2,2(R) given by a +c d a b T c d а — d

A:

Q: The domain of f(x) = V x2 + x – 6 | is : a) ]-4, 1[ b) ]-00, 4[U]1, +0[ c) ]-00,-3[U]2, +00이 d) ]-3,…

A:

Q: (3) Conjugate the map f(x) = 2x² +x – 3 to a map of the form Q(x) = x² + c. Find the conjugacy map h…

A:

Q: Let x0, x1, ..., Xn be distinct points on the interval [a, b], where Xo <x₂ < x₂ <--- < x₂ and let f…

A:

Q: Show that f: R\{2} → R\{5} defined by f(x) = (5x+1)/(x - 2) is bijective.

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: Let X = {a, b, c, d,e} and let T = {X, Ø, {a}, {a,d}, {a,e}}, then * O (X,T) is not connected but it…

A: No information is given for topology on Y So i have solved only the middle question

Q: 2c

A: The answer has been provided.Explanation:

Q: Suppose f(z) = z+z-2i 15 Define f(i) in such a way that the new function is . z +i continuous of…

A:

Q: b) Let f: D→ C be a function and Y a subset of C. Show that f¹ [C-Y] =D-f¹ [Y].

A:

Q: 3. Let x0, x₁,...,xn be distinct points on the interval [a, b], where Xo <x₁ < x₂ <... < xn and let…

A:

Q: Find all global minimum points of ƒ in R³, if ƒ (x1, x2, x3) = ex² + x½ + 2x²x3 + x² − 8x2 - 8x3

A:

Q: نقطة واحدة If f(z) = evsin 2z. Then 4 d Vsin 2z Vcot 2z. evain 2z %3D dz TO FO

A: ANSWER 2 :-

Q: (d) Let ø(x) = x² + x, x > 0. If defines an even extension of o to the entire real line, then for a…

A:

Q: Let g: RxR → RxR be a function defined by g(x, y) = (x², x-y). Suppose that X = Rx{0} and D= {(a, a)…

A: Bartleby Guideline : ''Since you have posted a question with multiple sub parts , we will provide…

Q: 3. The function f(m, n) = 2"3" is a one-to-one function from N x N into N. (a) Calculate f(m, n) for…

A:

Q: . Let a := V1+ v3. Find the minimal polynomial of a over Q.

A:

Q: Let P(x) = (x - 1)(x - 2)² (x-3)³. Find the set of r such that x (a) P(x) = 0, (b) P(x) 0.

A:

Q: 2 (a) Prove that, if a and b are nonnegative integers such that a 4 b, then 3ª = 3. (b) Using (a),…

A: Given a, b are non negative integers such that a≡4b . Then to show 3a≡53b. To show the function…

Q: f(x) = x² − 3 is the minimal polynomial of √3 over Q,

A: We have that              f(x) = x2-3  be a polynomial of two degree.

Q: Exercise 2. Let f : A → B be a function with Y1 and Y2 subsets of B. Prove that if Y1 C Y2, then…

A: Let f:A→B be a function and let Y1,Y2⊆B. Suppose that Y1⊆Y2 To show f-1(Y1)⊆f-1(Y2) Let x∈f-1(Y1)…

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.