Determine if the given functions from R3 to the given spaces

Determine if the given functions from R^3 to the given spaces are linear transformations. L([x, y, z]^T) = [-x + 5z, 2y]^T in R^2 Yes No L([x, y, z]^T) = [x + 1, y + 2, z + 3]^T in R^3 Yes No L([x, y,z]^T) = [Z - y, y - X, x - z]^T in R^3 Yes No

Solution

The matrix of first part will be of dimension 2X3

The constraints will give

ax + by + cz = -x + 5z

dx + ey + fz = 2y

Hence we get a=-1,b=d=f=0,c=5,e=2

Hence it is a linear transformation

Therefore correct answer is Yes

The matrix of second dimension will be of 3X3

Constraints are

ax + by + cz = x + 1

dx + ey + fz = y + 2

gx + hy + iz = z + 3

Not Linear Transformation, Answer is No

For the third part matrix will be of 3X3 and yes it is a linear transformation

a	b	c
d	e	f