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ilyas.h

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## Homework Statement

Let [itex]F_{2} = {0, 1}[/itex] denote a field with 2 elements.

Let V be a vector space over [itex]F_{2}[/itex]. Show that every non-empty set W of V which is closed under addition is a subspace of V.

## The Attempt at a Solution

subspace axioms: 0 elements, closed under scalar multiplication, closed under vector addition.

We can skip the latter axiom as it's given in the question.

proof of 0 element:

0, 1 ∈ F_2

x ∈ W

[itex]0x = 0_{W}[/itex]

therefore there exists a zero element.

proof of scalar multiplication:

0, 1 ∈ F_2

x ∈ W

[itex]1x = x[/itex]

this is true due to the scalar multiplication identity.

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I believe this could be wrong, I feel as though I am missing something. Thanks.

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