Understanding Vector Subspaces and Direct Sums

[franky2727]

probably not but if not i don't know where to start...
let U and W be vector subspaces of V. Show that UnW is a vector subspace of V
define the subspace U+W. What does it mean to say that V is a direct sum, UdirectsumW of U and W

understand the direct sum part but surely i need values of U and W to show UnW is a vector space of V?

 

[HallsofIvy]

The same way you show that any subset of a vector space is a subspace: show that it is closed under addition of vectors and scalar multiplication.

If u and v are vectors in UnW then, by definition of "intersection" they are in both U and W. Since u and v are both in U, and U is a subspace of V, u+ v is in U ...

Can you finish that?

 

[franky2727]

ye chears that helps a lot, could you take a look at my other post please? thanks

 

[franky2727]

hang on no i havn't read this properly at all, I'm still as stuck as i first was where has little v and u came from? if anything I'm more confused

 

[franky2727]

think I'm getting my head arround it now u and v elements of UnW so that u+v are elements of U and u+v are elements of W so because of this u+v are elements of UnW
the only thing I'm not getting is where this u and v are coming from? It would make more sense to me if it was u and w not u and v. and where is the w surely this is an element of UnW also?

i do however understand rules 1 and 3 multiplying by a scalar and zero vector just the addition that's baffeling me

 

[HallsofIvy]

You can call vectors anything you like! I wanted to avoid "u" and "w" specifically because someone might think that "u" must be in U and "w" must be in W.

Perhaps it would be better to say "if x and y are members of UnW then they are both members of U and both members of W. Since U is a subspace, it is closed under addition: x+ y is in U. Since W is a subspace, it is closed under addition: x+ y is in W. Since x+ y is in both U and W, it is UnW.

Now, you need to show that if x is in UnW, so is ax for a any number.

Oh, and you need to prove the subset is non-empty- most often that's done by showing that the 0 vector is in the set.

 

[franky2727]

thanks got it now