Finding an Orthogonal Base for Vector Space H

[Jimmy84]

Homework Statement

Greetings,

I'm trying to solve these problems

given the vectors u=(3,-2,1) and v=(2,-3,1)

1. Find an orthogonal base for the space H generated by {u,v}

2. Find the orthogonal projection of w=(3,0,1) on H

Homework Equations

The Attempt at a Solution

Im not sure how to get an orthogonal base.
Im thinking about finding a vector by inspection in which the dot product of v and u of that vector would be zero. Is there e method to find an orthogonal base for the space H ? Can I use gram-schmidt to solve the first problem and if so, how?

Thanks a lot for your comments.

 

[I like Serena]

Hi Jimmy84!

The Gram-Schmidt algorithm is exactly what you need.
Can you apply its steps?

When you have found an orthogonal base, it becomes easy to find an orthogonal projection.
Let's first try to apply Gram-Schmidt though...

What do you have about Gram-Schmidt and how much of it can you apply?

 

[Jimmy84]

Hi Jimmy84!

The Gram-Schmidt algorithm is exactly what you need.
Can you apply its steps?

When you have found an orthogonal base, it becomes easy to find an orthogonal projection.
Let's first try to apply Gram-Schmidt though...

What do you have about Gram-Schmidt and how much of it can you apply?

Hi!

My problems is that I am not sure how to apply Gram-Schmidt

given the vectors u=3,-2,1 and v=2 -3 1 I try to apply gram schmidt to them in this way the algorithm says that

w1 = u

then

w2 = v - (vu)/(uu) u

the result is w2 = -23/8, 1/4 ,-5/8 which is not orthogonal to u nor to v. I'm wondering what am I doing wrong?

 

[I like Serena]

You're on the right track! :)

But when I calculate w2 I get a different result.

What did you get for (vu)? And for (uu)?

 

[Jimmy84]

You're on the right track! :)

But when I calculate w2 I get a different result.

What did you get for (vu)? And for (uu)?

for vu I got 13 for uu 14 I think I found a mistake let me check the calculations again

 

[Jimmy84]

I found the mistake :) it is -11/14, -8/7 , 1/14 now can I say that w2 is an orthogonal basis of H? and how can I project w=(3,0,1) on H?

 

[I like Serena]

Yep!

Actually, a basis is a set.
So {w1, w2} is an orthogonal basis of H.

Do you have a formula for a projection?
(That formula is rather closely related to Gram-Schmidt.)

 

[Jimmy84]

I think that the formula for the projection of v into u is Proj u (v) = uv/(uu) u but that is for two vectors, I have no idea how to find the projection of a vector into a space.

 

[I like Serena]

It's the summation of the projections on the individual vectors that form the orthogonal basis.

In your case:
Proj H (x) = w1x/(w1w1) w1 + w2x/(w2w2) w2

 

[Jimmy84]

Thanks a lot, I was just wondering about a similar problem I'm trying to solve.

If there is a plane W with bases b1, b2 and an orthogonal base n , then

Proj W (b1) = b1 , Proj W (b2) = b2 , and Proj W (n) = 0 Im curious how can I compute Proj W (b1) = b1 ? Should I use the formula you just wrote for the plane W?

Thanks.

 

[I like Serena]

Yes, you can use that formula... can you apply it?

Also, you can take a look at the geometric meaning of it.
If you start with a vector in the plane you're projecting onto... you'll get that same vector!

 

[Jimmy84]

I'm have been for weeks trying to see the geometric meaning of it the problem is that this material is not covered in many books that I have consulted when it comes to reflection and orthogonal projection in planes. Can you recommend me a book that might touch the subject?

I'm reading anton howard`s elementary linear algebra.

I think the formula would be Proj W (b1) = (w1b1)/(w1w1) w1 + (w2b1)/(w2w2) w2

where w1 and w2 are an orthogonal basis for b1 and b2 isn't it?

 

[I like Serena]

Sorry, the books that I studied from are not available anymore.

Try w1=b1 and w2=b2.
And since b1 is orthogonal to b2, what is (b1b2)?

 

[Jimmy84]

so Proj W (b1) = b1b1/b1b1 b1 ? which is b1

Proj W (b2) = b2b2/b2b2 b2 = b2

I'm confused about the Proj W (n) how would it look like in order to compute it? my guess

is that it could be Proj W (n) = b1n/b1b1 b1 + b2n/ b2b2 b2

 

[I like Serena]

The projection using a normal vector n is:

Proj W (x) = x - (xn)/(nn) n

 

[Jimmy84]

Thanks a lot, have a good night.

 

[Jimmy84]

Sorry, the books that I studied from are not available anymore.

Try w1=b1 and w2=b2.
And since b1 is orthogonal to b2, what is (b1b2)?

Hi
well I was checking some details about the problem we were discussing yesterday in the example I am given b1 and b2 which are not orthogonal. Can the formula work if b1 and b2 are just non orthogonal bases of the plane W ?

 

[I like Serena]

Hi
well I was checking some details about the problem we were discussing yesterday in the example I am given b1 and b2 which are not orthogonal. Can the formula work if b1 and b2 are just non orthogonal bases of the plane W ?

Sorry, but no. b1 and b2 have to be orthogonal for the projection formula to work.
There's a reason why it's useful for a base to be orthogonal (or even better: orthonormal)!

 

[Jimmy84]

The problem is this one

1. Given a linear operator which can be an orthogonal projection or a reflection (passing through the origin) calculate the eigenvalues and eigenspaces associated which each eigenvalue.

The matrix is

2 1 -1
-1 0 1
1 1 0

2. based on that, determine if the operator is a reflection or an orthogonal projection.

3. Describe an orthogonal base of the plane, and complete it to a base of R^2. The matrix of the operator with respect to the calculated base must have the form

100
010
00-1

or

100
010
000

I got the eigenvalues 1 and 0 therefore I assume that it is an orthogonal projection.
the eigenvectors for the eigenvalue 1 are B { b1 = -1,1,0 and b2 = 1,0,1}

which were the ones I was thinking about applying in the formula that we were just discussing.

"Proj W (x) = w1x/(w1w1) w1 + w2x/(w2w2) w2 "In order to get the orthogonal projection matrix which vectors should I try in the formula of the Proj of W ?

im considering
Proj W (w1) = w1 , Proj W (w2) = w2 , where w1 and w2 are elements of the orthogonal basis of b1 and b2 however I'm not sure how to get the next column of zeros of the orthogonal matrix that I'm ask to find.

 

[I like Serena]

Good!

However, the projection formula only works for an orthogonal base.
If you want to use it, you will first have to make the base orthogonal.

 

[Jimmy84]

Im a bit confused,

Should I use Proj W (w3) = w1w3/w1w1 w1 + w2w3/w2w2 w2

for the last column of the orthogonal projection matrix ?

The first member is zero, but since w2 and w3 are not orthogonal it seems Proj W (w3) won't be zero?