Intersections of Subspaces and Addition of Subspaces

[TranscendArcu]

Homework Statement

http://img824.imageshack.us/img824/3849/screenshot20120122at124.png

The Attempt at a Solution

Let $S = \left\{ S_1,...,S_n \right\}$. If $L(S) = V$, then $T = \left\{ 0 \right\}$ and we are done because $S + T = V$. Suppose that $L(S) ≠ V$. Let $B_1 \in T$ such that $B_1 \notin L(S)$. Then the set $Q =\left\{ S_1,...,S_n,B_1 \right\}$ is linearly independent. If $L(Q) = V$ then we are done since $S + T = V$ and $S \cap T$ $= \left\{ 0 \right\}$. If not, then we may repeat the preceding argument beginning with the set Q. Thus, we would create a new set, call it Q', where $Q' = \left\{ S_1,...,S_n,B_1,B_2 \right\}$ and $B_1,B_2 \notin S$. Then, if $L(Q') = V$, then we are done. If not, then we continue as above. This process must certainly end because V is itself stated to be finite. Thus, such a T as in the problem must exist.

I think I'm going in the right direction, but I'm not sure if I'm executing the proof correctly.

 

[mighty2000]

Any help would be appreciated.

Hello,

Your proof seems to be on the right track. However, I think you may have some typos in your notation. Here are some suggestions for improving your proof:

1. Define S = {S1, ..., Sn} to be a set of vectors in a vector space V.

2. Since L(S) ≠ V, there exists a vector B1 ∈ V such that B1 ∉ L(S).

3. Consider the set Q = {S1, ..., Sn, B1}. Since B1 ∉ L(S), Q is linearly independent.

4. If L(Q) = V, then we are done since S + T = V and S ∩ T = {0}.

5. If L(Q) ≠ V, then there exists a vector B2 ∈ V such that B2 ∉ L(Q).

6. Consider the set Q' = {S1, ..., Sn, B1, B2}. Since B1, B2 ∉ L(Q), Q' is linearly independent.

7. If L(Q') = V, then we are done.

8. If L(Q') ≠ V, then we can continue this process, adding new vectors B3, B4, etc. until we have a set Q' = {S1, ..., Sn, B1, ..., Bk} such that L(Q') = V.

9. Since V is finite, this process must eventually end, and we will have a set T = {B1, ..., Bk} such that T ∩ S = {0} and S + T = V.

Overall, your proof seems to be on the right track, but you may want to clarify your notation and make sure to explicitly state each step in the proof. I hope this helps!