A linear transformation is defined by the formula Fx y z x

A linear transformation is defined by the formula F(x, y, z) = (x + 3y + z, -y + 3z). (a) Enter a basis of the linear transformation\'s kernel, as a list of vectors, for example [1, 2, 3], [4, 5, 6]. (b) Enter a basis of the linear transformation\'s range.

Solution

To find the kernel of the linear transformation T equate

T (x,y,z) = (0,0)

( x+3y+z, -y=3z) =(0,0) we get the eqns   

x+3y+z-0 , -y+3z=0 => y=3z and substituting in the 1 st eqn

x+9z+z=0 ie x= -10z   

Null space = { ( -10z ,3z,,z ) / whee z belongs to R }

Bsis of the nullspace = { ( -10 .3.1)} and nullity =1   

ii . to find the Range space find T (1,0,0) = (1,0)

T(0,1,0) = ( 1, -1)

T (0,0,1) = (0,3)

a=( 1,0) ,b=(1,-1) c=(0,3)   

we cansee that (0,3) = 9(1,0) -3(1,-1)

ie the 2 elements of the range space 91,0) , (1,-1) are linearly independent and form the basis of range space

asis of the Range space = { (1,0), (1,-1)} rank=2