Proving R(T) = Mn×n for T: Mn×n → Mn×n

[pyroknife]

Define T: M_n×n → M_n×n by T (A) = A^t . Show
that R(T) = M_n×n.

Alright then. The transformation is saying that it's transforming an nxn matrix into an nxn matrix (duh).

T(A)=A^t thus A^t is still an nxn matrix.

Thus R(T) is all of Mnxn?Is that it?

 

[pyroknife]

Also another part of the question is:
Define T: Mn×n → Mn×n by T (A) = A − A^t .
a. Find R(T ).
b. Find N(T ).a) R(T) is just the right hand side (A-A^t) which is the set of all skew symmetric matrices?

b) To find N(T) I set the RHS equal to 0. so A-A^t=0 thus A=A^t
so N(T) is the set of all symmetric matrices?

 

[Dick]

Yes for both. You seem to have a pretty good idea what you are doing. You might not need to post ALL of these questions.

 

[pyroknife]

Yes for both. You seem to have a pretty good idea what you are doing. You might not need to post ALL of these questions.

I'm insecure about most of my answers, even if a lot of them are right.