Subspaces and Orthogonality

[kdieffen]

Ok so I've been working on this problem and I'm really having some struggles grasping it. Here it is:

Let W be some subspace of Rn, let WW consist of those vectors in Rn that are orthognoal to all vectors in W.

1) Show that WW is a subspace of Rn?

So for this part I'm thinking that because WW is a linear combination of W (maybe) then therefore it forms a subspace of Rn

2) If {v1, v2,...vt} is a basis for W, show that a vector X in Rn lies in WW if and only if x is orthogonal to each of the vectors v1, v2,...vt?

And for this one I'm really at a lose for where to start. Any help would be appreciated.

Thanks

 

[Bacle]

What have you tried, so far.?

Basically, in order to show that any subset S of a vector space V is a subspace,

is to show that elements of S are closed under scaling and under linear combinations;

in this case, show that if w^ and w'^ are in W^ =(" W Perp" ) , so is their sum,

and for any w^, cw^ (c a scalar) is also in W^. The key here is the properties

of the inner-product.



For 2, one side follows by definition. For the other side ( w^ is perpendicular

to each of v1,..,vt ) , think of writing any x in W using the elements of v1,..,vt,

and, again, use the properties of the inner-product (multilinearity).

Good Luck.

 

[kdieffen]

1) Well I understand the definition of a subspace, i think I am just finding it difficult to fully understand the proof. Would it be correct to say then that:

WW is a subspace of Rn because
i) It contains the zero vector
ii) WW= {v1', v2',...vt'}

Let x, y span (WW)
x=av1' + av2' + avt'
y=bv1' + bv2' + bvt'

Therefore (x+y)=(a1+b1)v1' + (a2+b2)v2' + ...(at + bt)vt' and is closed under vector addition.

iii) Let x span (WW)

Then x= a1v1' + a2v2' + atvt' for some a1, a2,...at
Then kx=ka1v1' + ka2v2' + katvt'

Therefore it is closed under scalar multiplication. Since it satisfies all three, it is a subspace of Rn.

So that's what I have for part 1

And for part 2 I'm still lost lol

2)

 

[Bacle]

"" 1) Well I understand the definition of a subspace, i think I am just finding it difficult to fully understand the proof. Would it be correct to say then that:

WW is a subspace of Rn because
i) It contains the zero vector
ii) WW= {v1', v2',...vt'}

Let x, y span (WW)
x=av1' + av2' + avt'
y=bv1' + bv2' + bvt'

Therefore (x+y)=(a1+b1)v1' + (a2+b2)v2' + ...(at + bt)vt' and is closed under vector addition. ""


I don't see how you have shown it is closed. How do you know that x+y is in W^.?



iii) Let x span (WW)

What do you mean here.?. How do you know that a single vector x spans WW.?.
I think (reading below ) you mean: let x be a vector in WW. Right.?

Then x= a1v1' + a2v2' + atvt' for some a1, a2,...at
Then kx=ka1v1' + ka2v2' + katvt'

Therefore it is closed under scalar multiplication.

Not clear to me. How do you know kx is in WW.? You need to check this, or give a good
argument to that effect.

Since it satisfies all three, it is a subspace of Rn.

So that's what I have for part 1

And for part 2 I'm still lost lol.

Well, assume that a vector x in R^n is orthogonal to each of the basis vectors

{v1,..,vt} of W . You then want to show that x is orthogonal to any vector v in W.

How do you show any two vectors are orthogonal.?.

Assume x is orthogonal to each of v1,..,vt (what does this mean.?) and let w be in W.

What do you need to do to show that x and v are orthogonal.?. How can you write w in

order to show this.?

 

[blue_raver22]

Hello,

I understand that you are having some difficulty grasping the concept of subspaces and orthogonality. Let me try to explain it in a simpler way.

Firstly, a subspace of Rn is a subset of Rn that satisfies the three properties of a vector space - closure under addition, closure under scalar multiplication, and containing the zero vector. This means that if you take two vectors from the subspace and add them, the result will still be in the subspace, and if you multiply a vector from the subspace by a scalar, the result will also be in the subspace.

In this case, WW is a subspace of Rn because it satisfies these properties. If you take two vectors from WW and add them, the result will still be orthogonal to all vectors in W, and if you multiply a vector from WW by a scalar, the result will also be orthogonal to all vectors in W.

Now, for the second part, let's consider a vector X in Rn. If X lies in WW, it means that X is orthogonal to all vectors in W. This can be written as X⋅v1 = 0, X⋅v2 = 0, ..., X⋅vt = 0, where ⋅ represents the dot product. This is because the dot product of two orthogonal vectors is zero.

Conversely, if X is orthogonal to all vectors in W, it means that X⋅v1 = 0, X⋅v2 = 0, ..., X⋅vt = 0. This means that X lies in WW, as it satisfies the definition of WW.

I hope this helps you understand the concept better. Don't hesitate to ask for further clarification if needed. Keep up the good work!