If T V rightarrow W is a linear transformation from a vector

If T: V rightarrow W is a linear transformation from a vector space V to a vector space W then Ker(T) = {u epsilon V:T(u) = 0; the zero vector of W}. Prove the Ker(T) is a subspace of V. Prove that if the Ker(T) contains only the zero vector of V, then T is 1-1.

Solution

a)   To show that ker T is a subspace of V, we need to show that it has the following properties:

Clearly T(0)=0. So we need only show additive and scalar multiplicative closure.

Additive closure: We want to show that if a,bker(T) then a+bker(T)

For example, (a)=0,T(b)=0T(a+b)=0.

This follows from the property of additivity,

T(a+b)=T(a)+T(b) =0+0=0 and hence a+b ker T

Scalar multiplicative closure:

We want to show that if aker(T) then kaker T .

So T(a)=o and by the property of homogeneity (of degree 1) ,

we have that 0=k0=kT(a)=T(ka) and hence ka ker(T)

Therefore,

Ker(T) is subspace of V.