Prove Subspaces U & V in Rn: U \cap V is Subspace, U \cup V Not Necessarily

[danago]

Prove that if U and V are subspaces of Rⁿ, then so is U \cap V. Show by example that U \cup V is not necessarily a subspace of Rⁿ.

For the first part i can see that elements of U \cap V are elements of both subspaces U and V. If i take any two element from U \cap V and form some linear combination, then it must lie within U (since the elements are in U and U is a subspace) and it must also lie within V (since the elements are in V and V is a subspace), hence, U \cap V is also a subspace.

Thats what i would have called informal reasoning. How can i actually prove that this is the case? Is my reasoning enough to stand as a solid proof?

For the second part i didnt have any troubles. I just defined:

<br /> \begin{array}{l}<br /> U = \{ k\left( {\begin{array}{*{20}c}<br /> 1 \\<br /> 1 \\<br /> \end{array}} \right)|k \in R\} \\ <br /> V = \{ k\left( {\begin{array}{*{20}c}<br /> 1 \\<br /> 0 \\<br /> \end{array}} \right)|k \in R\} \\ <br /> U \cup V = \{ a\left( {\begin{array}{*{20}c}<br /> 1 \\<br /> 1 \\<br /> \end{array}} \right),b\left( {\begin{array}{*{20}c}<br /> 1 \\<br /> 0 \\<br /> \end{array}} \right)|a,b \in R\} \\ <br /> \end{array}<br />

I then chose two elements from the union of U and V and formed a linear combination which was not an element of the union of U and V, hence U and V is not a subspace.

<br /> \begin{array}{l}<br /> \left( {\begin{array}{*{20}c}<br /> 0 \\<br /> 1 \\<br /> \end{array}} \right),\left( {\begin{array}{*{20}c}<br /> 1 \\<br /> 1 \\<br /> \end{array}} \right) \in U \cup V \\ <br /> \left( {\begin{array}{*{20}c}<br /> 0 \\<br /> 1 \\<br /> \end{array}} \right) + \left( {\begin{array}{*{20}c}<br /> 1 \\<br /> 1 \\<br /> \end{array}} \right) = \left( {\begin{array}{*{20}c}<br /> 1 \\<br /> 2 \\<br /> \end{array}} \right) \notin U \cup V \\ <br /> \end{array}<br />

I think that's the way to go about it, so its really just the first part i needed a bit of help with.

Thanks,
Dan.

 

[Defennder]

Well you can write it in this form:

Suppose \vec{u_1},\vec{u_2} \in U \cap V.

Clearly that implies that \vec{u_1} \ \mbox{and} \ \vec{u_2} \in U and this in turn implies that for any a,b \in \Re , a\vec{u_1} + b\vec{u_2} \in U since U is a subspace of R^n. What can you say about u1,u2 in relation to V now? And how would you relate this to U \cap V?

 

[n_bourbaki]

Prove that if U and V are subspaces of Rⁿ, then so is U \cap V. Show by example that U \cup V is not necessarily a subspace of Rⁿ.

For the first part i can see that elements of U \cap V are elements of both subspaces U and V. If i take any two element from U \cap V and form some linear combination, then it must lie within U (since the elements are in U and U is a subspace) and it must also lie within V (since the elements are in V and V is a subspace), hence, U \cap V is also a subspace.

Thats what i would have called informal reasoning. How can i actually prove that this is the case? Is my reasoning enough to stand as a solid proof?

Yes. It is a perfectly good formal proof. In fact I would much rather see a simple explanation like that than some string of symbols like u,v,a,b with decoration.

 

[danago]

cheers for the replies guys. All pretty clear now :)