Question

For each of the following pairs of ordered bases β and β′ for P₂​(R), find the change of coordinate matrix that changes β′-coordinates into β-coordinates.

(a) β={x₂,x,1} and β′={a₂​x₂+a₁x+a₀,b₂​x₂+b₁x+b₀,c₂x₂+c₁x+c₀}
(b) β={1,x,x₂} and β′={a₂x₂+a₁x+a₀,b₂x₂+b₁x+b₀,c₂​x₂+c₁x+c₀​}
(c) β={2x₂−x,3x₂+1,x₂} and β′={1,x,x₂}
(d) β={x₂−x+1,x+1,x₂+1} and β′={x₂+x+4,4x₂−3x+2,2x₂+3}
(e) β={x₂−x,x₂+1,x−1} and β′={5x₂−2x−3,−2x₂+5x+5,2x₂−x−3}
(f) β={2x₂−x+1,x₂+3x−2,−x₂+2x+1} and β′={9x−9,x₂+21x−2,3x₂+5x+2}

Answer 1

To convert coordinates from one base to another in P2(R), express each vector from the new base as a linear combination of the old base vectors. Use the coefficients from these expressions as columns to form the change of coordinate matrix.

Step-by-step explanation:

To find the change of coordinate matrix that changes β'-coordinates into β-coordinates for polynomials in P₂(R), we must express each vector in the base β' in terms of the basis β. This is done by writing each vector of β' as a linear combination of vectors in β and using the coefficients from these linear combinations to fill the columns of the change of coordinate matrix.

The process generally involves solving systems of equations, with the goal being to write:

1. β'[1] = aβ[1] + bβ[2] + cβ[3]
2. β'[2] = dβ[1] + eβ[2] + fβ[3]
3. β'[3] = gβ[1] + hβ[2] + iβ[3]

The coefficients a, b, c, d, e, f, g, h, i will form the matrix:

[ [a, d, g], [b, e, h], [c, f, i] ]