Which of the following transformations are linear transforma

Which of the following transformations are linear transformations? If the given transformation is a linear transformation, write down its standard matrix. T(x, y) = (3x - y, x, 2x + y) T(x, y) = (x, 2y) T(x,y) = (y, x,x + 1) T(x,y) = (0, 1, 0) T(x,y,z) = (0, 0) T(x,y,z) = (y, 0, z2) T(x, y, z) = (3y, x + 2y,3x - z) T(x, y, z) = (5x - 7y, 2x + z, 0)

Solution

Solved first four sub-parts, post one more question to get the remaining answers. Thanks

a) The first transformation is al linear transformation, the resulting vectors is 2 X 1 , hence the initial vector will be of the size 2 X 2

The elements of the matrix A will be equal

ax + by = 3x - y

cx + dy = 2x + y

Hence a = 3, b= -1, c=2 and d=2

e) This transformation is al linear transformation, the resulting vectors is 2 X 1 , hence the initial vector will be of the size 2 X 3

The elements of the matrix A will be equal

all the elements of this matrix will be equal to 0

b) The first transformation is al linear transformation, the resulting vectors is 2 X 1 , hence the initial vector will be of the size 2 X 2

The elements of the matrix A will be equal

ax + by = x

cx + dy = 2y

Hence a = 1, b= 0, c=0 and d=2

f) This transformation is non-linear, hence the standard matrix is not possible since it contains the term z^2

a	b
c	d