Question

 Consider the basis S = {v1, v2} for R2, where v1=(1,1) and v2=(2,3)and 

T: R2🡪P2 be the linear transformation such that T (v1)= 2-3x+x2 and T(v2)=1-x2. Find the formula for T and use that to find T(a, b).

Answer 1

Let $u_1=(1,0)$ and $u_2=(0,1)$. Then

$T(v_1) = 2-3x+x^2$

$T(v_2)=1-x^2$

$T(u_2) = T(v_2-2v_1) = T(v_2)-2T(v_1) =(1-x^2) - 2(2-3x+x^2) = x^2 + 6x -3$

$T(u_1) = T(v_1-u_2) = T(v_1) - T(u_2) = (2-3x+x^2)-(x^2 + 6x -3) = 5-9x$

$T(a,b) = T(au_1+bu_2) = aT(u_1)+bT(u_2) = a(5-9x) + b(x^2 + 6x -3) = bx^2 + (6b-9a)x +(5a-3b)$

Answer. $T(a,b) = bx^2 + (6b-9a)x +(5a-3b)$