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## Homework Statement

Hi I'm trying to prove that the sum of two subspaces [tex]U[/tex] and [tex]W[/tex] is also a subspace.

## Homework Equations

[tex]U[/tex] is a subspace of [tex]V[/tex] if [tex]U[/tex] is also a vector space and it contains the additive identity, is closed under addition, and closed under scalar multiplication.

The definition of a sum a vector subspace U and W is

[tex]U+W=\{u+w:\, u\in U,w\in W\}[/tex]

## The Attempt at a Solution

1. Since [tex]U[/tex] and [tex]W[/tex] both contain the additive identity, [tex]U+W[/tex] contains the additive identity

3. Since both [tex]U[/tex] and [tex]W[/tex] are closed under scalar multiplication, any combination of [tex]u+w[/tex] is closed under scalar multiplication since multiplication is distributive, associative and commutes (assuming were dealing with the reals here).

I'm having a hard time thinking about how to justify that U+W is closed under addition.

Also is my justification for closure under scalar multiplication right?

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