Showing that a set of linear algebra statements are true

[trap101]

V is the direct sum of W₁ and W₂ in symbols:

V = W₁ (+) W₂ if:

V = W₁ + W₂ and
W₁ $\cap$ W₂ = {0}

Show that the following statements are equivalent:

1. V = W₁ (+) W₂

2. Every vector v$\in$V can be written uniquely as w₁ + w₂ where w₁$\in$W₁ and w₂$\in$W₂

3. V = W₁ + W₂ and for vectors w₁$\in$W₁ and w₂$\in$W₂ if w₁ +w₂ = 0 then w₁ = w₂

4. If $\alpha$₁ is a basis for W₁ and $\alpha$₂ is a basis for W₂ then
$\alpha$ = $\alpha$₁ $\cup$ $\alpha$₂

Attempt:

I'm thrown off here because I'm given the definition of a direct sum. So how can I even show that all these statements are equivalent? Would I start by assuming one of the statements is true? For example: Let's say statement 1 is true. that means...well it means exactly what statement 2 is. Maybe take a vector in W₁ and one in W₂ sum them together and create a vector in V? But then to do that I would have to define an exact vector space...

 

[Mark44]

V is the direct sum of W₁ and W₂ in symbols:

V = W₁ (+) W₂ if:

V = W₁ + W₂ and
W₁ $\cap$ W₂ = {0}

Show that the following statements are equivalent:

1. V = W₁ (+) W₂

2. Every vector v$\in$V can be written uniquely as w₁ + w₂ where w₁$\in$W₁ and w₂$\in$W₂

3. V = W₁ + W₂ and for vectors w₁$\in$W₁ and w₂$\in$W₂ if w₁ +w₂ = 0 then w₁ = w₂

4. If $\alpha$₁ is a basis for W₁ and $\alpha$₂ is a basis for W₂ then
$\alpha$ = $\alpha$₁ $\cup$ $\alpha$₂

Attempt:

I'm thrown off here because I'm given the definition of a direct sum. So how can I even show that all these statements are equivalent? Would I start by assuming one of the statements is true? For example: Let's say statement 1 is true. that means...well it means exactly what statement 2 is. Maybe take a vector in W₁ and one in W₂ sum them together and create a vector in V? But then to do that I would have to define an exact vector space...

Start by assuming that statement 1 is true. Then show that this implies that statement 2 must be true, which you can do by invoking the definition of a direct sum.

Next show that statement 2 being true implies that statement 3 is true. Then show that statement 3 being true implies that statement 4 is true. Finally, show that statement 4 implies statement 1.

You don't have to define a vector space. You don't mention it, but it seems to me that V is some arbitrary vector space and that W₁ and W₂ are subspaces of V.

 

[trap101]

Start by assuming that statement 1 is true. Then show that this implies that statement 2 must be true, which you can do by invoking the definition of a direct sum.

Next show that statement 2 being true implies that statement 3 is true. Then show that statement 3 being true implies that statement 4 is true. Finally, show that statement 4 implies statement 1.

You don't have to define a vector space. You don't mention it, but it seems to me that V is some arbitrary vector space and that W₁ and W₂ are subspaces of V.

But then what is it that I'm really doing beyond repeating the definition of a direct sum? Because it seems that those each just logically follow each other. Yes it is suppose to be an arbitrary vector space.

 

[Dick]

But then what is it that I'm really doing beyond repeating the definition of a direct sum? Because it seems that those each just logically follow each other. Yes it is suppose to be an arbitrary vector space.

Yes, they do follow from each other. That's exactly what you are supposed to show. Each statement is different from the definition of direct sum. You have to show why they are equivalent.

 

[trap101]

Yes, they do follow from each other. That's exactly what you are supposed to show. Each statement is different from the definition of direct sum. You have to show why they are equivalent.

Ok, I'm still puzzled at how I can show it. Because if I didn't have the 4 statements for example. Starting with the definition of a direct sum, I would've just said that the definintion of a direct sum implies that every v$\in$ V can be written uniquely as a combination of w1 and w2. I don't see how I'm "showing" this though.

 

[Dick]

Ok, I'm still puzzled at how I can show it. Because if I didn't have the 4 statements for example. Starting with the definition of a direct sum, I would've just said that the definintion of a direct sum implies that every v$\in$ V can be written uniquely as a combination of w1 and w2. I don't see how I'm "showing" this though.

Show HOW the implication works. You start with v=w1+w2=w1'+w2'. What property of the direct sum let's you say w1=w1' and w2=w2'? This isn't a hard proof by any means, but there is something to say.

 

[trap101]

Show HOW the implication works. You start with v=w1+w2=w1'+w2'. What property of the direct sum let's you say w1=w1' and w2=w2'? This isn't a hard proof by any means, but there is something to say.

Well the only properties I have are that the intersection of the two sets is {0}. Also each vector in the subspace is unique. I suppose by uniqueness:

there exists a unique vector in w1 and w2 s.t v is the combination of them?...Still not seeing it.

 

[Dick]

Well the only properties I have are that the intersection of the two sets is {0}. Also each vector in the subspace is unique. I suppose by uniqueness:

there exists a unique vector in w1 and w2 s.t v is the combination of them?...Still not seeing it.

You don't HAVE uniqueness, you want to SHOW uniqueness. If v=w1+w2=w1'+w2' (w1, w1' in W1 and w2, w2' in W2) then w1-w1'=w2'-w2. So?

 

[trap101]

You don't HAVE uniqueness, you want to SHOW uniqueness. If v=w1+w2=w1'+w2' (w1, w1' in W1 and w2, w2' in W2) then w1-w1'=w2'-w2. So?

I'm going to go out on a limb and say 0=0, but I don't think that would make sense considering I'm trying to show uniqueness so I can't assume w1-w1' = 0 and the same for w2. And so my struggle continues with this sort of thing being my impediment from obtaining A's. sigh.

 

[Dick]

I'm going to go out on a limb and say 0=0, but I don't think that would make sense considering I'm trying to show uniqueness so I can't assume w1-w1' = 0 and the same for w2. And so my struggle continues with this sort of thing being my impediment from obtaining A's. sigh.

Which subspace is w1-w1' in? Same question for w2-w2'.

 

[trap101]

Which subspace is w1-w1' in? Same question for w2-w2'.

w1-w1' is in W1 and w2-w2' is in W2. ==> W1 = W2 but that doesn't do anything.

unless:

v = (w1-w1') + (w2 - w2')

 

[Dick]

w1-w1' is in W1 and w2-w2' is in W2. ==> W1 = W2 but that doesn't do anything.

unless:

v = (w1-w1') + (w2 - w2')

So call u=w1-w1'=w2'-w2. From what you have said that must mean u is in W1 and W2. What's the only vector that's both in W1 and W2?

 

[trap101]

So call u=w1-w1'=w2'-w2. From what you have said that must mean u is in W1 and W2. What's the only vector that's both in W1 and W2?

the only vector in W1 and W2 is the 0 vector. I'm still puzzled as to how this shows statement 2.

 

[Dick]

the only vector in W1 and W2 is the 0 vector. I'm still puzzled as to how this shows statement 2.

So u=0. Then w1-w1'=0 and w2'-w2=0. Remember what you want to prove?

 

[trap101]

So u=0. Then w1-w1'=0 and w2'-w2=0. Remember what you want to prove?

As far as I remember I'm trying to prove statement 2. Then with that I'll prove statement 3, etc. But the fact that u = 0, how is this showing statement 2? I'm trying to go over the steps but I'm failing to see the connection

 

[Dick]

As far as I remember I'm trying to prove statement 2. Then with that I'll prove statement 3, etc. But the fact that u = 0, how is this showing statement 2? I'm trying to go over the steps but I'm failing to see the connection

Don't worry about the parts yet. You are trying to show 1)->2). You want to show that if v=w1+w2=w1'+w2', then w1=w1' and w2=w2'. I think you are practically there is if you string all the stuff we've been doing together.