Resolved: "a) Prove that the set β={p0(x)p1(x),…,pn(x)} of Lagrange..."

Reggie

Answered question

2021-09-05

a) Prove that the set

β = {p_{0} (x) p_{1} (x), \dots, p_{n} (x)}

of Lagrange polynomials is linearly independent in

P_{n} (b)

. Deduce that

β

basis for

P_{n}

diskusje5

Skilled2021-09-06Added 82 answers

Given:

β = {p_{0} (x) p_{1} (x), \dots, p_{n} (x)}

To Prove
(a). Prove that the set ẞ of Lagrange polynomials is linearly independent in Pn.
(b). Deduce that

β

is a basis for

P_{n}

.
a) let

c_{0}, c_{1}, \dots, c

are constants

\Rightarrow c_{0} p_{0} (x) + c_{1} p_{1} (x) + c_{2} p_{2} (x) + \dots + c_{n} p_{n} (x) = 0

Put

x = x_{0}

in (i)

c_{0} p_{0} (x_{0}) + c_{1} p_{1} (x_{0}) + c_{2} p_{2} (x_{0}) + \dots + c_{n} p_{n} (x_{0}) = 0

\Rightarrow c_{0} \times 1 + c_{1} \times 0 + \dots + c_{n} \times 0 = 0

\Rightarrow c_{0} = 0

Continuing (

x = x_{1}, x = x_{2}, x = x_{3}, \dots, x = x_{n}

) we get

c_{1}, c_{2}, \dots, c_{n}

must be zero

\Rightarrow p_{0}, p_{1}, p_{2}, \dots, p_{n}

are linearly independent
b) (

p_{0}, p_{1}, p_{2}, \dots, p_{n}

) is a basis of

P_{n}

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