Definition to find the adjoint of given matrix

Question

Problems:

Find a 5x5 matrix M>0 such that if a = [α_i]⁴_i=0, b = [β_i]⁴_i=0  ∈ R⁵ andx(t)= ∑⁴_k=0 α_kt_k, y(t) = ∑⁴_k=0  β_kt^k  then (x,y)_v = b'Ma

Can we use this definition to find the adjoint of T (T is given at the end)?

This part is the additional information to solve the question above;

Let V=L2[-1,1] be the Hilbert space of functions over the time interval [-1,1] with inner product

(x,y)_v = ∫¹_-1x(t)y(t)dt

Let P5 ⊂ V be the subspace of polynomials of order 4 or less, endowed with the inner product and norm of V, and let X = {x^k(t) = t^k}^4k=0  ⊂ P⁵ , be its natural basis. The linear transformation S is defined as

   (Sx)(t) = x(t) =d/dt x(t)=d²/dt²x(t)=1.5 ∫¹_-1r²x(r)dr

T is defined as the restriction of S to P5.

(sx)(t) = x(t)=x(t)=x(t)=1.5 ∫¹_-1r²x(r)dr = (2a₀+a₁+13/5 a₂+3/7a₄)+(a₁+2a₂+6a₃)+t+(a₂+3a₃+12a₄)t²+(a₃+4a₄)t³+a₄t⁴ ∈ P⁵

SX=TX

The matrix representation of T is below;