The function T 1P2 rightarrow 1P2 given by Ta bx cx2 a b

The function T: 1P_2 rightarrow 1P_2 given by T(a + bx + cx^2) = (a - b) + (b - c)x + (c - a) x^2 is a linear transformation Evaluate T(-1 + 6x + 2x^2) = (-1 -6) + (6 -3)x + (2 +1)x^2 = -7 + 4x + 3x^2 Determine whether the following are linear transformation either prove that it is or prove it is not T: 1R^3 rightarrow 1R^1 given by T[x y z] = 3x T: 1R^3 rightarrow 1R^2 given by T[x y z] = [x^2 + y z - x] T(n + v) = T(u)+T(v) T(k n) hT(n)

Solution

1) T(a+bx+cx²)=(a-b)+(b-c)x+(c-a)x² is linear transformation then

T(-1+6x+2x²) =(-1-6)+(6-2)x+(2-(-1))x² =-7+4x+3x²

2) T(x,y,z) R³

=T(3X,0Y,0Z) =T(3X) is a linear transformation in R²

T (x,y,z)=T(x²+y,z-x) is not a linear transformation,since addition and scalar multiplications can be done in linear transformation and nothing more