Question

Question 1: (4+4 =8 Marks) [a] Show that the transformation 7(x, y) = (7x - 3y: 5x - 2y) of R4 R4 is a linear and give the matrix representation "A" of T with respect to the standard basis B={(1,0),0,1)). Furthermore, prove that T is invertible and find the preimage of the vector (1,-4). [b] Consider the transformation T: P3 → Pz defined by Tax3 + bx? +cx+d) = (a +2d)x? +(6+20)x² +(a+c+d)x. Determine Ker(T) and Range(T); and find a basis for all them. Verify the rank nullity theorem.

Answer 1

Now express I in column matrin form (13) = (1-3y) (Sx-zy , To find the images of the vectors in the standard basis. +(1:1) = 70-360) (5(1-2(0) - (3) And +((:)) = (16)-369] - 1:3) Now, the standard matix is 7 - 3 Art el S - (Al = 7(-2) – 5(-3) = -14+15 = 1 0 Hence À exist.

The preimage of the vertor (1,-4) is T' (1,-4) = (-261) +3(<4), -S()+7(-4)) - (-2-12, -5-28) =(-14, -33)