Question

Find a basis {p(x),q(x)} for the vector space {f(x)∈P₃ [x]∣f (4)=f(1)} where P₃ [x] is the vector space of polynomials in x with degree less than 3 . p(x)=,q(x)=

Answer 1

To find a basis for the vector space f(4)=f(1), we need to find two polynomials, p(x) and q(x), that satisfy the given condition. It involves solving a system of equations.

Step-by-step explanation:

To find a basis for the vector space f(4)=f(1), we need to find two polynomials, p(x) and q(x), that satisfy the given condition.

Let's start by considering the general form of a polynomial in P₃[x]: f(x) = ax₂ + bx + c. Since f(4) = f(1), we can substitute these values into the equation to get:

a(4)₂ + b(4) + c = a(1)₂ + b(1) + c

This simplifies to 16a + 4b + c = a + b + c. Now we have a system of equations: 15a + 3b = 0. We can solve this system to find the values of a and b, which will give us a basis.