Linear algebra image and kernels Let fR3 R2 with fxyzxyzxyz

Linear algebra image and kernels

Let f:R³ ->R² with f(x,y,z)=(x+y-z,x+y-z)

a) find a basis for im (f)

b) find a basis for ker(f)

c) check your answers to (a) and (b) agree with dimension theorem

Solution

a) Let (x,y,z) in R³ such that:

f(x,y,z) = (x+y-z,x+y-z) = x(1,1)+y(1,1)-z(1,1) = (x+y-z)(1,1)

hence basis for Im(f) = (1,1)

b) Let (x,y,z) in Ker(f)

f(x,y,z) = (x+y-z,x+y-z) = (0,0)

=> x+y-z=0

=> x+y=z

Therefore, (x,y,z) = (x,y,x+y) = x(1,0,1)+y(0,1,1)

Therefore the basis of Kert(f) = {(1,0,1), (0,1,1)}

c) By dimension theorem:

din(Im(f))+dim(ker(f)) = dim(R³)

Now dim(Im(f)) = 1 and dim(Ker(f)) = 2 (Dimesion equals number of vectors in the basis)

Also dim(R³) = 3

Hence we get that dim(Im(f))+dim(Ker(f)) = dim(R³)