If V is any vector space and S and T are linear operators on V such that ST=TS show that the null space and the range of T are invariant under S.
I think I need to begin by taking an element of the range of T and having S act on it and show that it stays in V? Can you help get me started?
S(x1 + x2) = S(x1) + S(x2) (this is because S is a linear map)
= T(y1) + T(y2) therefore (x1 +x2) in M
and
S(ax) = aS(x) = aT(y) therefore (ax) in M
I think this is better (I hope)! Thanks you've been a great help!
S(x1 + x2) = S(x1) + S(x2) (this is because S is a linear map)
= T(y1) + T(y2) = M(x1) + M(x2)
and
S(ax) = aS(x) = aT(y) = aM(x)
Thanks for your help, is this kind of right?
S,T: V onto W are both linear maps. Show that M:={x out of V s.t. Sx out of Range(T)} is a subspace of V
I know that to show M is a subspace of V I must show:
i. 0 out of M
ii. For every u, v out of M, u+v out M
iii. For every u out of M, a out of F, au out of M.
I just don't know...