V=R² is a vector space and T is a linear operator. T=(2x-y, x+y). f(x)=2+3x and g(x)=x+x² . a) find f(T) and g(T) linear operators. b) Are these operators singular?

Question

c) Find dimensions of ker f(T), ker g(T), rank f(T), rank g(T)

Answer 1

Matrix A of linear transformation: T(1, 0) = (2, 1); T(0, 1) = ( -1, 1). So, matrix of linear transformation is

A = [(2, 1) (-1, 1)].

a) So f(A) = 2I +3A = 2[(1, 0) (0,1] + 3[(2, 1) (-1,1)] = [(8, 3) (-3, 5)] ;

Now g(A) = A + A² = [(2, 1) (-1, 1)] + [(2, 1) (-1, 1)]².= [(2, 1) (-1, 1)] + [(3, 3) (-3, 0)] = [(5, 4) (- 4, 1)].

b) f(A) nonsingular operator because detf(A) = 40+ 9 = 49 ≠ 0

g(A) also nonsingular operator because detg(A) = 5 + 16 = 21 ≠ 0