Question

Let T:ℙ2(ℝ)→ℙ2(ℝ) be a linear transformation given by T(f(x))=3f′(x)+9f(x). If TS:ℝ3→ℝ3 is the corresponding coordinate transformation with respect to the standard basis for P2, {1,x,x2}, compute the matrix AS of the coordinate transformation. (Hint: Consider how T transforms an arbitrary polynomial of the form f(x)=a+bx+cx2.) AS= ⎡⎣⎢⎢⎢⎢⎢ ⎤⎦⎥⎥⎥⎥⎥

Answer 1

An arbitrary polynomial in P₂( R) is of the form f(x) = a+bx+cx² , where a,b,c are arbitrary real numbers. Then f’(x) = b+2cx so that T(f(x) = 3(b+2cx)+ 9(a+bx+cx²) = (9a+3b)+(9b+6c)x+9cx².

Further, {1,x,x²} is the standard basis for P₂( R) and T(1) = 9, T(x) = 3+9x and T(x²) = 6x+9x².

Thus, the matrix AS of the corresponding coordinate transformation TS: R³ → R³ is given by AS =

9	3	0
0	9	6
0	0	9

It may be observed that the entries in the columns of AS are the   scalar multiples of 1 and the coefficients of x,x² in T(1),T(x), T(x²) respectively.