proof about subspace

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[grave monolith]

Here's my proof for (a)(->): Since W is a subspace of V, W is a vector space with the operations defined on V. Therefore, 0 is in W by VS 3.

the proof in text is longer so I'm wondering if it's valid

[hoary basin]

That's the wrong direction.

What you're actually saying would be a proof that W being a subspace would imply condition a.

[grave monolith]

yeah that's what I'm saying but why is that the wrong direction?

[hoary basin]

Because the forward implication is that a subset of V which satisfies the three conditions is a subspace of V.

[grave monolith]

ok if it’s backward then is the proof valid?

[austere snow]

@grave monolith What's your definition of vector subspace?

I'm under the impression that people just learn what you showed in the theorem as a definition

[austere snow]

Yeah ok I see, then

[austere snow]

I think there's something missing sorta?

like, say the 0 of V is denoted 0_V

And the 0 of W is denoted 0_W

there is essentially no reason why they should be the same thing at this stage yet

The point (a) argues that 0_V is in W, which you have yet to prove

You know that 0_W is in W, but not that 0_V is in W

(it should be noted that for a vector space V, the 0_V given by VS 3 is provably unique, which has important consequences)

[grave monolith]

How do I know it’s who’s 0 in (a)?

[austere snow]

First of all, did you prove uniqueness of the zero?

in a vector space

[grave monolith]

Let’s say there’s 01 and 02, then x+01=x=x+02, and by theorem 1.1 in the book 01=02, so 0 is unique

[austere snow]

Nice, thanks for that

So this makes things a whole lot easier

you just have to verify that 0_V is a neutral element of W, as in, it is a good candidate to be a 0_W

by uniqueness, it is the 0_W

[grave monolith]

I’m not sure what neutral element means, does that imply I have to show 0_v + x=x for all x in W?

[grave monolith]

In general, do we need to say that what 0 being mentioned?

[austere snow]

The origin, the zero

[austere snow]

So you know W has a unique element 0_W that verifies x + 0_W = 0_W + x = x for all x in W

But you don't yet know whether or not 0_W = 0_V

But you can prove that 0_V is also in W and verifies the same properties

And by uniqueness, 0_V = 0_W

[austere snow]

Yeah it's ambiguous here

[austere snow]

Why would 0_V be in there?

[grave monolith]

Because W is a subset of V, so for every x in W, 0_V+x=x holds, so 0_V is a zero element for W

[austere snow]

0_V is indeed a zero element for W, but this does not yet guarantee that 0_V is in W

[austere snow]

Here's another hint: take any element in W, say w

What is w + (-1)w?

[grave monolith]

But doesn’t W have a zero element?

[austere snow]

It does, it's a vector space and verifies VS 3

You know:

• 0_V is in V
• 0_W is in W
• 0_V verifies 0_V + x = x + 0_V = x for all x in W

You don't know, but must prove one of the two following:

• 0_V is in W
• 0_V = 0_W

(Note that here, if you can prove one of the two, the other one follows automatically)

[grave monolith]

If I prove the zero element for w is unique, then 0_V = 0_W?

[austere snow]

You already know it's unique

But if 0_V is in W, it is a fortiori a zero element in W, therefore it is the zero element of W

conversely, if 0_V = 0_W, it is obvious that 0_V is in W

so you have to prove that one of the two holds, at least

[grave monolith]

But 0_V and 0_W are both zero element for W, and also zero is unique, what am I missing?

[austere snow]

You can imagine a situation where 0_W is not equal to 0_V

so it is a nonzero element of V that acts like a zero only on W

Such a situation cannot happen, indeed, but it has not yet been proven why

[hoary basin]

To be clear, you have not yet proved why. Lest you believe there are unproven mathematical statements which are accepted as fact.

[grave monolith]

Doesn’t that mean we have 2 zeros for W?

[hoary basin]

Why? If 0_W exists such that for all x in W, x + 0_W = x, but there exists y in V such that y + 0_W =/= y, then you could have 0_W in W and 0_V not in W.

[austere snow]

No, the zero in W is unique, but it may not necessarily be the zero as in 0_V until proven so

[hoary basin]

Hypothetically, unless you can prove we can't.

[austere snow]

nothing says that the red dot should be exactly the black dot

(It is true that a situation where 0_W is different from 0_V cannot happen, but you must show why)

[grave monolith]

When I prove uniqueness of zero, I used theorem 1.1, which assumes the vectors are from the same vector space

[grave monolith]

0_W

[austere snow]

On the one hand, yes

But on the other hand, w is also a vector in V since W is a subset of V

So?

[grave monolith]

also equals to 0_V

[austere snow]

Conclusion?

[grave monolith]

0_V = 0_W so 0_V is in W

[austere snow]

Good job

[grave monolith]

Thx a lot

[hoary basin]

Yes, and 0_W is unique in W.

[grave monolith]

+close

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