If T is a linear operator on R^n , show that the Ker(T) and Rng(T) are invarient under T.?

Answer 1

(i) T(Ker(T)) = {0}, which is contained in Ker(T).
(ii) T(Rng(T)) is contained in T(R^n), which equals Rng(T).

Answer 2

Andrew's answer is fine. Alternatively you can use an element driven approach:

1) let v be an element of Ker(T), then Tv=0 which is an element of Ker(T) by linearity of T.

2) Let v be an element of Rng(T) then Tv is an element of Rng(T) by virtue of the fact that T is a function from a set to itself.

But we really are just saying the same thing.