# Subspaces in Vector Spaces over F2

• karnten07
In summary: Is 1 the only multiplicative identity element then?Great, thanks for clearing that up.So since 0 and 1 are the only scalars used on V, then i could simultaneously prove it is closed under scalar multiplication and show the existence of a multiplicative identity element (1).Yes, 1 is the only multiplicative identity element.
karnten07

## Homework Statement

Let (F2) ={0,1} denote the field with 2 elements.

i) Let V be a vector space over (F2) . Show that every non empty subset W of V which is closed under addition is a subspace of V.

ii) Write down all subsets of the vector space (F2)^2 over (F2) and underline those subsets which are subspaces.

## The Attempt at a Solution

For i.) do i need to show that it is closed under scalar multiplication also? I don't understand how it is because for example 6x1=6 which is not of F2??

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To show that W is a subspace of V, it suffices to show that for two scalars in your field, a and b, and two vectors in W, v and w, av + bw is in W.

6 x 1 = 6 is certainly not in F2, but it is in V (assuming V is R).

I think your confusion is that you're letting V = F2.

e(ho0n3 said:
To show that W is a subspace of V, it suffices to show that for two scalars in your field, a and b, and two vectors in W, v and w, av + bw is in W.

6 x 1 = 6 is certainly not in F2, but it is in V (assuming V is R).

I think your confusion is that you're letting V = F2.

Yes you are right, that's what i was kind of thinking. Could anyone explain to me what it means that V is a vector space over F2? please.

For part ii.) i have that F2^2 = {(0,0),(0,1),(1,0),(1,1)}

So i assume that 0,0 is a subspace because it is the zero subspace. Do i say that (0,1) is a subspace also as this is the vector space of F2 itself?? Are (1,0) and (1,1) subspaces also?

karnten07 said:
Yes you are right, that's what i was kind of thinking. Could anyone explain to me what it means that V is a vector space over F2? please.

Oh does it mean that the scalars that can be applied are only the two elements 0 and 1 of F2?

e(ho0n3 said:
It seems to me that you have very little knowledge of what vector spaces are. Perhaps this will help: http://en.wikipedia.org/wiki/Vector_space.

Yes, i have read about vector spaces and it does seem to me that in this case the vector space is over F2 and F2 only consists of 2 elements. So does this mean only 0 and 1 are the scalars that this vector space deals with, so to speak??

Yes, 0 and 1 are the only allowed scalars. As long as F2 is a field, the vector space over it is well-defined.

e(ho0n3 said:
Yes, 0 and 1 are the only allowed scalars. As long as F2 is a field, the vector space over it is well-defined.

Great, thanks for clearing that up.

So since 0 and 1 are the only scalars used on V, then i could simultaneously prove it is closed under scalar multiplication and show the existence of a multiplicative identity element (1).

## 1. What is a subspace?

A subspace is a subset of a vector space that is closed under addition and scalar multiplication. This means that for any two vectors in the subspace, their sum and any scalar multiple of the vector is also in the subspace.

## 2. How do you determine if something is a subspace?

To determine if something is a subspace, you must check if it satisfies the three properties of closure under addition, closure under scalar multiplication, and contains the zero vector. If all three properties are met, then it is a subspace.

## 3. What is the difference between a subspace and a vector space?

A subspace is a subset of a vector space, while a vector space is a set of vectors that can be added and multiplied by scalars. A subspace must also satisfy the three properties mentioned above, while a vector space does not necessarily have to.

## 4. Can a subspace be empty?

No, a subspace cannot be empty. It must contain at least the zero vector to satisfy the closure under addition and scalar multiplication properties. If a subset does not contain the zero vector, it cannot be a subspace.

## 5. Is the intersection of two subspaces also a subspace?

Yes, the intersection of two subspaces is also a subspace. This is because it will still satisfy the three properties of closure under addition, closure under scalar multiplication, and contains the zero vector.

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