Write the matrix [T]G which represents T with respect to the basis G The basis G

Question

Write the matrix [T]G which represents T with respect to the basis G

The basis G = {1 + x, 1 - x, 1 - x2, 1 - x3}

T(ax3 + bx2 + cx + d) = (a - b)x2 + (c - d)x + (a + b - c)

Explanation / Answer

T(ax3 + bx2 + cx + d) = (a - b)x2 + (c - d)x + (a + b - c) = u(1+x)+v(1-x)+w(1-x^2)+z(1-x^3) =>(a-b)x^2+(c-d)x+(a+b-c) = -zx^3-wx^2+(u-v)x+(u+v+w+z) Comparing terms we get, -z = 0 => z = 0 -w = a-b u-v = c-d u+v+w+z = a+b-c =>w = b-a u+v = a+b-c+a-b = 2a-c u-v = c-d =>2u = 2a-d =>u = a-0.5d v = u-c+d = a-0.5d-c+d = u-c+0.5d Therefore T(ax3 + bx2 + cx + d) = (a - b)x2 + (c - d)x + (a + b - c) = (a-0.5d)(1+x)+(u-c+0.5d)(1-x)+(b-a)(1-x^2)+0(1-x^3)

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