Question

Show that the transformation T defined by T(X), x)) = (2x - 3X2, X, +4,6x) is not linear. If T is a linear transformation, then T(0) = and T(cu + dv) = CT(u) + dT(v) for all vectors u, v in the domain of T and all scalars c, d.

Linear algebra

Answer 1

Given the Transformation is defined as -

T21,22) = 2:01 – 3.02,11 +4, 6:02)

Let x and y be two vectors in the domain of T , then T is a linear Transformation if -

(i) T(x+y)=T(x)+T(y)

(ii) Tar) = aT()

Let we have -

, AND ,

x = (21.02) and y = (y1, y2)

(1+y) = (11,12 + y1, y2) = (11 + 91,12 + y2)

T21,22) = 2:01 – 3.02,11 +4, 6:02)

T(41, y2) = (2y1 – 3y2, 41 +4, 6y2)

Therefore ,

T2+ y) = 211 + y1) – 312 + y2), (01 + y1) + 4,612 + y2)

= 2.01 + 2y1 – 3.02 - 3y2, 11 +41 +4,602 + 6y2

Rearranging the terms in above expreesion keeping all terms of x and y separate to each other

= 2:01 – 3.12, 11 +4, 6:02) + (2y1 – 3.12, 41, 642)

T 1+y) = 2.11 – 3.12, 11 +4,6x2) + (2y1 – 3.62, 41, 6y2)

T 1+y) =111, 12) + (2y1 – 3.62, 41, 6y2)

T(x+y) = (2) + (2y1 – 3.02, 41, 642)

Thus , clearly ,

T (1+y) #T(2) +T(y)

Therefore ,

Given transformation T is not linear .

If T is a linear transformation , then this transformation will always take the zero vector to zero .

Therefore , If T is a linear transformation , then T(0) = 0.