1. The problem statement, all variables and given/known data Let U and W be subspaces of V. What are U+U, U+V? Is U+W=W+U? 2. Relevant equations 3. 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.