Consider the linear transformation defined by x' = ax + by

MATHEMATICS · Subjective · CBSE · Mathematics · Matrices · Matrices

Problem

Consider the linear transformation defined by

x' = ax + by, \quad y' = cx + dy,

where

a,b,c,d

are real constants. (i) For a triangle with vertices

(x_1,y_1)

(x_2,y_2)

and

(x_3,y_3)

, express the area of the image triangle in terms of

a,b,c,d

and the coordinates of the original vertices. (ii) Hence show that the area of the image triangle is

|ad - bc|

times the area of the original triangle. (iii) Verify the result obtained in part (ii) for the specific transformation

x' = x + 2y,\; y' = 3x + 4y

applied to the triangle with vertices

(0,0)

(1,0)

and

(0,1)