What Are the Rank and Nullity of a Linear Transformation?

[mathmathmad]

Homework Statement

find the rank and nullity of the linear transformation T:U -> V and find the basis of the kernel and the image of T

Homework Equations

U=R[x]<=5 V=R[x]<=5 (polynomials of degree at most 5 over R), T(f)=f'''' (4th derivative)

The Attempt at a Solution

Rank = 2
Nullity = 4

basis of kernel = {1,x,x^2,x^3} ?
since a kernel is mapped to V, then the image is the zero vectors? and the basis of the image of T is the empty set?

 

[radou]

The basis for Im(T) can be {1, x}.

 

[mathmathmad]

another QUESTION

T : U -> V

and the kernel(T) is the zero vectors,
then what is the basis? it's not the empty set?

 

[radou]

If it happens, for some linear transformation T, that ker(T) = {0} holds, then yes, the basis for ket(T) is the empty set.