Gram-Schmidt orthonormal basis

If you need to find v_2, you need to take the inner product of u_2 with v_1, which is why we have the inner products in the first place. The formula for v_2 is given right there.In summary, to convert the given vectors u1=(1,1,1), u2 = (1,1,0), u3 = (1,0,0) into a normal orthonormal basis using the Gram-Schmidt process, we first normalize u1 by dividing it by its length which is found using the dot product formula. Then, using the general formula for Gram-Schmidt, we find v2 by taking the inner product of u2 with v1,
  • #1
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Homework Statement


Let R^3 have the inner product <u, v> = u1v1 + 2u2v2 + 3u3v3. Use the Gram-Schmidt process to convert u1=(1,1,1), u2 = (1,1,0), u3 = (1,0,0) into a normal orthonormal basis


Homework Equations


I know the process for the orthonomoral converasion. I have no problem with that. I hope I can skip typing those equations out.

The Attempt at a Solution


My confusion is how to use the equation for the dot product. I am thinking that I need to do the Gram-Schmidt process then somehow put the result into the dot product equation.

If I phrased anything wrong or left out something pertinent I apologize this is my first post. So I tried to follow the rules.
 
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  • #2
If you know the formula, then what is the problem? You have the vectors and the inner product, so you just plug and chug.
Let's start you off - we need to normalize u_1, i.e replace u_1 with

u_1/ sqrt(<u_1,u_1>) = u_1/sqrt(6).

Note that this is the least sensible way to do this. It would be better to use u_3 as the first element to orthonormalize, since that satisfies <u_3,u_3> = 1, i.e. it is already a unit vector. But I presume that the book/class wants you to go through them in order and normalize u_1 to get v_1, then to work out u_2 - <u_2,u_1> and normalize that, etc.
 
  • #3
Then maybe I am reading too far into. I am just not sure how the dot product equation in the beginning of question is utilized. As in after I convert [tex]\vec{u_1}[/tex], [tex]\vec{u_2}[/tex], [tex]\vec{u_3}[/tex]. So do I just plug in the new normalized [tex]\vec{u_1}[/tex], [tex]\vec{u_2}[/tex], [tex]\vec{u_3}[/tex] and then use [tex]\vec{u_1}[/tex] = [tex]\vec{v_1}[/tex], [tex]\vec{u_2}[/tex]=[tex]\vec{v_2}[/tex], [tex]\vec{u_3}[/tex]=[tex]\vec{v_3}[/tex] into <u, v> = u1v1 + 2u2v2 + 3u3v3.
 
  • #4
The general Gramm-Schmidt formula reads:

[tex]v_j=a_j-\left( \frac{\langle a_j,v_1 \rangle}{\langle v_1,v_1 \rangle}v_1+\frac{\langle a_j,v_2 \rangle}{\langle v_2,v_2 \rangle}v_2+\ldots+\frac{\langle a_j,v_{j-1} \rangle}{\langle v_{j-1},v_{j-1} \rangle}v_{j-1} \right)[/tex].

This should make it pretty obvious where to use the dot product, so I am curious what kind of formula were you thinking of?
 
  • #5
For example, to "normalize" the first vector, you need to divide by its length. And its length is [itex]||v||= \sqrt{v\cdot v}[/itex]. Because of the definition of dot product, [itex](1,1,1)\cdot(1,1,1)= 1(1)+ 2(1)(1)+ 3(1)(1)= 6[/itex]. That's where matt grime got the [itex]\sqrt{6}[/itex] he divided the first vector by.
 
  • #6
Ok, I was forgetting/missing to use the equation when getting the norm or mag to normalize at the end. I think I am good. Thanks for all the quick help.
 
  • #7
You don't just use it at the end to normalize - you use it at every occasion that you need to take an inner product.
 

1. What is Gram-Schmidt orthonormal basis?

Gram-Schmidt orthonormal basis is a method used in linear algebra to convert a set of linearly independent vectors into a set of orthonormal vectors. This process involves taking a set of vectors and creating a new set of orthonormal vectors that have the same span as the original set.

2. Why is Gram-Schmidt orthonormal basis important?

This method is important because it allows us to find a more convenient and efficient set of basis vectors for a vector space. Orthonormal vectors have many useful properties, such as being easier to work with in calculations and providing a geometric interpretation of vector operations.

3. How does the Gram-Schmidt process work?

The Gram-Schmidt process involves a series of steps to transform a set of linearly independent vectors into a set of orthonormal vectors. First, a vector is selected as the first basis vector. Then, each subsequent vector is projected onto the previous basis vectors and subtracted from itself to create a new vector. This new vector is then normalized to create an orthonormal vector. This process is repeated for each vector in the original set.

4. Are there any limitations to using Gram-Schmidt orthonormal basis?

One limitation of this method is that it can be computationally expensive for large sets of vectors. Additionally, if the original set of vectors is not linearly independent, the resulting orthonormal set may not accurately span the same space as the original set.

5. In what applications is Gram-Schmidt orthonormal basis used?

This method is commonly used in fields such as engineering, physics, and computer science. It is particularly useful in applications involving vector spaces, such as signal processing, image compression, and data analysis.

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