Determine the range and Kernel of transformation Ta b c dbbx

Determine the range and Kernel of transformation T[a b c d]=b+bx+bx^2

Solution

T : V W, where V is a subspace of R⁴ and W is the set of all polynomials, of degree upto 2, of the form b + bx + bx² where b R.

The kernel of T is the set of all vectors v in V such that T (v) = 0. Let v = [a, b, c, d] be an arbitrary vector in V. Then T(v) = T( [a, b, c, d ]) = b + bx + bx² . Now, T(v) = 0, when b + bx + bx² = 0, i.e. when b = 0. Thus, if v = ( [a,0,c,d ]), then T(v) = 0. Then Ker (T) = { [a,0,c,d] : a, c, d R}.

The range of the linear transformation T is the set of all vectors w in W such that there is a vector v in V with T(v) = w. Let w = b + bx + bx² be an arbitrary vector in W. Then T (v) = w if v = [ a, b, c, d ]. Thus the range of T is the set of all polynomials of the form a + bx + cx² such that a = c = b and b R.