Question

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

Answer 1

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

$p(x)=a_o+a_1x+a_2x^2+.....+a_nx^n$

where $a_0,..,a_n$ 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-

$p(x)=a_o+a_1x+a_2x^2+.....+a_nx^n$

$q(x)=b_o+b_1x+b_2x^2+.....+b_nx^n$

Then The addition operation is given by-

$p(x)+q(x)=a_o+b_o+(a_1+b_1)x+(a_2+b_2)x^2+.....+(a_n+b_n)x^n$

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.