Sums of Subspaces: U+U, U+V, Is U+W=W+U?

[cookiesyum]

Homework Statement

Let U and W be subspaces of V. What are U+U, U+V? Is U+W=W+U?

Homework Equations

The Attempt at a Solution

It is easy to show that U+U and U+V are spaces too under closed addition and scalar multiplication, but I'm not sure where they lie. For example, is U+U a subspace of V or V+V. The space U+V confuses me altogether; does it lie outside of V? And for the last one, does the question mean the direct sum of U + W even if the plus doesn't have a little circle around it? If it is a direct sum, wouldn't the statement be true because U and W together form the basis of V? I'm just having a really difficult time picturing all of this.

 

[mighty2000]

Dear fellow scientist,

Thank you for your question. Let me try to provide some clarification on the concepts of subspaces and direct sums.

First off, U+U and U+V are indeed subspaces of V, as you have correctly stated. They are subspaces because, by definition, a subspace is a subset of a vector space that is closed under addition and scalar multiplication. Since U+U and U+V satisfy these conditions, they are subspaces of V.

Now, to answer your question about where these subspaces lie, it is important to understand that a subspace of a vector space is also a vector space in its own right. Therefore, U+U and U+V are both subspaces of V, but they are not necessarily the same as V or each other. They can have different dimensions, bases, and elements, but they still satisfy the conditions of a vector space.

Moving on to the concept of direct sums, the notation U+W does indeed refer to the direct sum of U and W. This means that every element in U+W can be uniquely written as a sum of an element from U and an element from W. This is different from the sum U+V, where the elements in U and V do not necessarily have to be unique. So, in this case, U+W and W+U are the same because the order of the subspaces does not matter in a direct sum.

I hope this helps to clarify things for you. If you have any further questions, please do not hesitate to ask.