Show that the given mapping is a linear transformation FILL

Show that the given mapping is a linear transformation (FILL IN THE BLANKS).

defined by

T(x₁, x₂) = (x₁ + 7x₂, 7x₁ x₂).

Let

Thus, T is a linear transformation.

T((x1, x2) + (y1, y2))	=	T(x1 + y1, x2 + y2)
	=
	=	(x1 + 7x2, 7x1 x2) +
	=	T(x1, x2) + T(y1, y2)

Solution

T (x₁+y₁,x₂+y₂)= (x₁+y₁+7x₂+7y₂, 7x₁+7y₁-x₂-y₂)= (x₁+7x₂,7x₁-x₂)+(y₁+7y₂,7y₁-y₂)= T(x₁,x₂)+T(y₁,y₂)

T(cx₁,cx₂)= (cx₁ +7cx₂, 7cx₁-cx₂)= c(x₁+7x₂,7x₁-x₂) = cT(x₁,x₂)