In the vector space P2, of polynomials of degree less than or equal to 2, with real coefficients, consider the inner product defined by [image] and the norm associated with it, that is, ∥p∥ =...

In the vector space P2, of polynomials of degree less than or equal to 2, with real coefficients, consider the inner product defined by
[image]
and the norm associated with it, that is, ∥p∥ = √⟨p(t),p(t)⟩.

Remember that a vector is said to be unitary if its norm is equal to one. In relation to this inner product, answer the following items, mathematically justifying your statements.

(a)
Determine the value of the scalar k so that the polynomials p(t)=t+k8 and q(t)=−t2 are orthogonal.

(b)
The vector subspace W ⊂ P₂
generated by the vectors p(t)=4t−3 and q(t)=−t2 is the same generated by the polynomials of the previous item. Obtain a basis for W where the vectors are orthogonal and unitary.