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

In summary: Your Name]In summary, the conversation discussed the concepts of subspaces and direct sums in a vector space. It was clarified that U+U and U+V are both subspaces of V, but they may not be the same as V or each other. The notation U+W refers to the direct sum of U and W, where every element can be uniquely written as a sum of an element from U and an element from W. The order of the subspaces does not matter in a direct sum.

## Homework Statement

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

## 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.

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.

## 1. What is the definition of "sum of subspaces"?

The sum of two subspaces U and V is the set of all possible combinations of vectors from both subspaces. In other words, it is the set of all vectors that can be written as a sum of a vector from U and a vector from V.

## 2. How is the sum of two subspaces U and V denoted?

The sum of two subspaces U and V is denoted as U+V.

## 3. What is the difference between U+U and U+V?

U+U refers to the sum of two copies of the same subspace U, while U+V refers to the sum of two different subspaces U and V. U+U may result in a smaller subspace than U+V, as it only includes combinations of vectors from one subspace.

## 4. What does the equation U+W=W+U mean in terms of sums of subspaces?

This equation means that the sum of two subspaces U and W is equal to the sum of two subspaces W and U. In other words, the order in which we add subspaces does not matter when calculating the sum.

## 5. How do we determine if U+W=W+U?

To determine if U+W=W+U, we can check if every vector in U+W can also be found in W+U, and vice versa. If this is true, then U+W=W+U and the equation holds.

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