Question

Find a basis for each of the spaces V in Exercises 16 through 36 , and determine its dimension. 25. The space of all polynomials f(t) in P₂

​such that f(1)=0

Answer 1

To find a basis for the space of polynomials in P₂ with f(1)=0, we substitute 1 for t in the polynomial and simplify the equation. The basis for this space is {(1, -1, 0), (0, 1, -1)}.

Step-by-step explanation:

To find a basis for the space of all polynomials in P₂ such that f(1)=0, we need to find the polynomials that satisfy this condition.

Let's consider a polynomial f(t) = at² + bt + c. Since f(1) = 0, we can substitute 1 for t in the polynomial: a(1)² + b(1) + c = 0. Simplifying this equation gives us a + b + c = 0. This means that any polynomial in P₂ that satisfies this equation can be a basis for the space. Therefore, a basis for this space can be {(1, -1, 0), (0, 1, -1)}.