Question #169302

If On denotes the vector space of all polynomial of degree ≤ n , give two linearly independent elements of P4/PE.


1
Expert's answer
2021-03-12T05:42:26-0500

A Polynomial of degree n is a function of the form-


p(x)=ao+a1x+a2x2+.....+anxnp(x)=a_o+a_1x+a_2x^2+.....+a_nx^n


where a0,..,ana_0,..,a_n are real numbes called coefficient and n is called the degree of the polynomial.


Polynomial can be represented by the vector of their coefficient in vector space as-


Linear operation are defined on the piolynomial of same degree i.e n


Consider two polynomial-

p(x)=ao+a1x+a2x2+.....+anxnp(x)=a_o+a_1x+a_2x^2+.....+a_nx^n


q(x)=bo+b1x+b2x2+.....+bnxnq(x)=b_o+b_1x+b_2x^2+.....+b_nx^n


Then The addition operation is given by-

p(x)+q(x)=ao+bo+(a1+b1)x+(a2+b2)x2+.....+(an+bn)xnp(x)+q(x)=a_o+b_o+(a_1+b_1)x+(a_2+b_2)x^2+.....+(a_n+b_n)x^n


Here polynomial p(x) and (g(x) are linearly independent since niether the value of P(x) depends upon Q

nor G(x) value depens upon P.


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