- #1

tylerc1991

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## Homework Statement

Suppose [itex]U[/itex] is a subspace of [itex]V[/itex]. What is [itex]U + U[/itex]?

## Homework Equations

There are two definitions of a subspace sum that I know of (the first is the definition given in my book):

(1) [itex]U_1 + U_2 = \{ u_1 + u_2 : u_1 \in U_1, u_2 \in U_2 \}[/itex]

(2) [itex]U_1 + U_2 = \text{ span} ( U_1 \, \bigcup \, U_2)[/itex]

## The Attempt at a Solution

Before I tried to solve the general problem, I thought about a specific example. Suppose that [itex]U = \{ (x,y) \in \mathbb{R}^2 : y = x \} \subseteq \mathbb{R}^2[/itex]. Now [itex]U + U = U[/itex]. So I should expect the same answer in general.

Using the first definition of a subspace sum:

[itex]U + U = \{ u + u : u \in U \} = \{ 2u : u \in U \}[/itex]

Using the second definition of a subspace sum:

[itex]U + U = \text{ span} (U \, \bigcup \, U) = \text{ span}(U) = \{ au : a \in \mathbb{F}, u \in U \}[/itex].

While these are very similar(one is a special case of the other), I am leaning towards the second answer. That being said, how could I have come up with the second answer using the first definition of a subspace sum?