Question

Let T be a linear operator on a vector space V and let w1, w2, w3...... wk be T-invariant subspaces of V. prove that w1+ w2......wk is also a T-invariant subspace of V

Answer 1

T(w1+w2 + ... wk) = T(w1) + T(w2) + ... +T(wk)

Answer 2

could you please elaborate.. i got the point but still not completely

Answer 3

so, first show that w1+w2 + ... + wk is a subspace by showing it satisfies the 3 properties of a subspace, call it S
1. 0 is an element of S
2. if x1 and x2 are in S then x1+x2 is in S
3. if x1 is in S and c is a scalar, then cx1 is in S

then show that T(w1+w2+...+wk)=T(w1)+T(w2)+ ... +T(w3)
thus w1+w2+...+wk is invariant under T.

Answer 4

1. because w1, w2, ..., wk are all subspaces, then 0 is in each of these. thus 0 is in S = w1+w2+...+wk = 0+0+...+0 = 0.

Answer 5

get the idea?

Answer 6

ya i have proved the first part i.e w1, w2.....wk is a subspace.. have problem in showing T(w1+w2..wk)= T(w1)+ T(w2)+......T(wk)

Answer 7

T is a linear operator, therefore T(u+v) = T(u) + T(v)

Answer 8

oh.. thanx a lot.. such a small detail i missed out..thank you.

Answer 9

you're welcome!