Problem 5 Let R be the field of real numbers and {e₁,e2} be the standard basis for R². Consider the linear operator 7 € L(R²) defined by T(е₁) = ₁ + 2e₂ 7(e₂)=4e₁ +3e2 Find the minimal polynomial for 7 and show that the rational canonical form for 7 is R R = What are the elementary divisors of T. B = Hint: Given TEL(V), if m,(x) = ao+a₁x + +ak is the minimal polynomial for T then = {v, 7(v),..., 7k-¹(v)} is a basis for V and [T]B is the companion matrix of T and gives the minimal polynomial. Now, consider v=e₁

Q: Determine whether {2x³ - 3x², 3x+5, 5+x³, 4x − x²} C P (3) is a basis for P(3).

A: 2x3-3x2,3x+5, 5+x3,4x-x2⊂P(3)

Q: The set B=(3+2x², 12+2x+8x²,- (29+6x+20x2)) is a basis for P2. Find the coordinates of p(x) = 8 + 4x…

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Q: 3. Find the ranks and nullities of the following linear maps T: U → V, and find bases of the kernel…

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Q: The set B = {4 — 4x², 12 - x - 12x², 33- 3x - 32x² } is a basis for P₂. Find the coordinates of p(x)…

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Q: EX 7 Using the scalar form of the table A /1 3 5 2 3 1 1-α 1 6 4-α -2 12 6 + a 2 -5 19 to determine…

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Q: 1. Let B₁ = = {-1} {[]} i) Find Rep ii) Find Rep B1 and -{[]} B2 = be two ordered bases for R² C…

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Q: -1 -2 -3 -4- 1 3 Let A = 1 1 1) Compute det (A) using the cofactor expansion along the fi 2) Find…

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you.  If…

Q: Let V = P2(R). Show that 3 = {1+r+2²,x+r², x²} is a basis for V. [You may assume that we already…

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Q: Problem 7. Prove or disprove: There exists a basis po, P1, P2, P3 of P3(R) such that none of these…

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Q: Please

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Q: Consider a basis C= {[3] [3]} Let (41c = [-2] What is in standard coordinates? of Rn

A: Given- The basis is C=03,33 of Rn and x→C=−2−2. To find- The x→ in standard coordinates.

Q: Define T:R-→R² by T(x) = Ax. Find a basis B for R² with the property that [T]B is diagonal. - 3 - 3…

A: Given:T:ℝ2→ℝ2 is defined by Tx=AxWhere A=-3-3-1-5To find basis B for ℝ2 such that TB is diagonal.…

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Q: 4. Find the Lagrange and Newton interpolate polynomials P3(x) of nodes (-1,3,5), (1,-5.5), (2,-1),…

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Q: You are given that B = {1 – x², 1+ x, 2x + 3x²} is a basis for P2 and - 2 [p(x)]B= |3. If p(x) = a +…

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Q: b-a a. Let A = then A is diagonalizable. Select one: True False

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Q: 11. For what value(s) of k does the following set of vectors form a basis for R³? ·4·0 3 3

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Q: Consider the following ordered basis for P₂ (R): B = {x² + 2x + 1, x + 1, 1} (a) The quadratic…

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Q: Which one of the following is an orthonormal basis of R³? {Q.0·0} 1 1 امام امامت امام 0 3 0 L √2 0…

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Q: (a) (b) (c) Let W be the subpace of M2x2 (R) b W = { [20 + $ 30+d]:akedER} a, b, c, Find a basis for…

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Q: 4[X]- = 9x₁x₂. representation P for L relative to B₁ and B₂ such that [L(u)]₂=P[u]₁ Let L: R2 R2 be…

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Q: Problem 1. A. Prove that B = {1+x, x+x?,1+x²) is a basis for P2 second degree polynomial space

A: Basis: Let V be a vector space, then a set of vector B={V1,V2,...,Vn} is called basis of set V is…

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Q: 0555 if A = 0 0 0 0 then the basis fqf 0 205 N(A") is 1 .a Hạmmad Selawe .b كواے 1 .c 0.

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Q: -6 Let B ={[31],[3]}andc={[3],[4]} 2 C be bases for R². B If [P] = [ 2 ]; [P]c?

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Q: Question 3. Create a basis B for P3 that contains exactly 2 elements from the set A = {1+x + x³, 1…

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Q: Suppose that 0.3210 approximates 0.3245. How many significant digits does the approximate have? O 3…

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Q: The set B = {3 − 3x², 15+ 4x − 15x², 48x² - (47+ 12x)} - - is a basis for P2. p(x) = 12x² [p(x)] B…

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Q: basis: The set B = {-(4+x²), x − 12 − 3x², 49 − 3x + 12x²} is a basis for P2. Find the coordinates…

A: Let the coordinates of p(x) = 4x - 40 - 10x2 w.r.t the given basis B be (a, b, c)t  Then,      4x -…

Q: - Suppose that f(x) = x² + 1, g(x) = x² - x and h(x) = x − 1. Then the set of functions {f, g, h} is…

A: DETAILED SOLUTION IS PROVIDED BELOWExplanation:

Q: Use the standard basis for P2, B = {1, t, t²}, and the coordinate mapping to show that 10 – 12t –…

A: The solution is below.

Q: Find the degree and a basis for Q(√3 + √5) over Q(√15). Findthe degree and a basis for Q(√2, 3√2,…

A: In this question, we have to find the degree and a basis for Q(√3 + √5) over Q(√15) and the degree…

Q: See picture

A: Does the set L of two vectors form a basis for the subspace S?