Thread: direct sum of subspaces

Hey, I am confused about this question. I need some help getting started.

Let where is a vector space and are the subspaces of .
Use the fact that to prove that is the direct sum of the subspaces .

Where should I start with this? Thanks for any help.

EDIT: sorry made a mistake in my working.

Does anyone have any ideas on how to prove it?

Thanks

you are given a clue as to where to start (a fact to use). since i am not an unkind person i wll give you another: what is dim(U1 + U2)?

what can you say therefore about dim((U1+U2)+U3),....,dim(U1+U2+....+Uj)?

Originally Posted by Deveno

you are given a clue as to where to start (a fact to use). since i am not an unkind person i wll give you another: what is dim(U1 + U2)?

what can you say therefore about dim((U1+U2)+U3),....,dim(U1+U2+....+Uj)?

So


Thus and .

Is this correct? Where to from here?

Thanks

if dim(V) = n, what must any n-dimensional subspace be?

Originally Posted by Deveno

if dim(V) = n, what must any n-dimensional subspace be?

If we let be a subspace of then and if the .

So any n-dimensional subspace must be .

So in relation to my question, therefore is the direct sum of the subspace .

Does this complete the proof?

Thanks for your help Deveno!