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Let f-1, fo, and f₁ denote the Lagrange polynomials associated with -1,0, and 1 respectively. (2.1) Find f1, fo, and f, and express each one in standard polynomial form, that is, a + bx + cx² where a, b, and c are real numbers. (2.2) Show that f₁(-x) = f(x) and f-₁(-x) = f(x). (2.3) Show that fo and f1 + f₁ are even. (2.4) Show f = P₂ (R) is even if and only if f = span{fo, f_₁ + f₁}.

Let f-1, fo, and f₁ denote the Lagrange polynomials associated with -1,0, and 1 respectively. (2.1) Find f1, fo, and f, and express each one in standard polynomial form, that is, a + bx + cx² where a, b, and c are real numbers. (2.2) Show that f₁(-x) = f(x) and f-₁(-x) = f(x). (2.3) Show that fo and f1 + f₁ are even. (2.4) Show f = P₂ (R) is even if and only if f = span{fo, f_₁ + f₁}.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 59EQ
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QUESTION 2 Let f-1, fo, and f₁ denote the Lagrange polynomials associated with -1,0, and 1 respectively. (2.1) Find f1, fo, and f₁ and express each one in standard polynomial form, that is, a + bx + cx² where a, b, and c are real numbers. (2.2) Show that f₁(-x) = f(x) and f-1(-x) = f(x). (2.3) Show that fo and f1 + f₁ are even. (2.4) Show f = P₂(R) is even if and only if f = span{fo, f₁ + f₁}.
Transcribed Image Text:QUESTION 2 Let f-1, fo, and f₁ denote the Lagrange polynomials associated with -1,0, and 1 respectively. (2.1) Find f1, fo, and f₁ and express each one in standard polynomial form, that is, a + bx + cx² where a, b, and c are real numbers. (2.2) Show that f₁(-x) = f(x) and f-1(-x) = f(x). (2.3) Show that fo and f1 + f₁ are even. (2.4) Show f = P₂(R) is even if and only if f = span{fo, f₁ + f₁}.
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