Use the Gram-Schmidt process to transform the basis {u1,u2,u3} into an orthonormal basis

Question

Let R3 have the Euclidean inner product and

 

u1= [2]       u2= [1 ]    u3= [4 ]

       [-2]            [-1]           [-3]                    or it can be written as u1(2,-2,1),  u2(1,-1,5) and u3 (4,-3,4)

       [1 ]            [5 ]            [4]                                                                                                                                             

 

Use the Gram-Schmidt process to transform the basis {u1,u2,u3} into an orthonormal basis

 

Answer 1

The first step would be to check if the three vectors are linearly independent. Well, in this problem, it's given that they are a basis, so they must be. But, this is not always how Gramm Scmidt problems are given and the process will not work on a linearly dependent set. An easy way to check is to organize them into a 3x3 matrix and take the determinant of that matrix. If the determinant is not zero, then the vectors are linearly independent. 

 

Now, all you have to do is choose which vector from these three will be the first vector in your new basis and apply the Gramm Scmidt process. This process will give you an orthogonal basis for the three dimensional space. Then, for the set to become orthonormal all of the vectors will have to be orthogonal to each other and have a magnitude of 1. So, all you have to do is scale the vectors you obtained from the process. It is very easy to do this. In each of the vectors, divide each element by the vector's magnitude. The new vectors should all have a magnitude of one and because we only normalized the the length of the vectors so that they all have length one, we have not changed their directions and they remain mutually orthogonal. Hence, this set will be our answer.

 

In sum, check if the set is linearly independent. The Gramm Schmidt process will take a set of n linearly independent vectors and output a set of n orthogonal vectors that span the n- dimensional space. Essentially, it will yield a basis composed of orthogonal vectors. The only difference between that and an orthonormal basis is that all of the vectors in an orthonormal basis have a magnitude of one. So, we "normalize" each vector's magnitude by dividing the elements in each vector by the the magnitude of the vector they are a part of. Then you are done.

 

I am sorry, if you were looking for direct calculations, but the Gramm Schmidt process is fairly self explanatory. If the explicit calculations are what is alluding you then go back and look up "dot product" which is what I believe they are referring to by Euclidean Inner Product. The definition of orthogonality, linear independence, basis for a vector space and an orthornormal set. After you know those things, the answer will look like it's already on the page. The Gramm Schmidt process merely takes a basis and orients it so the vectors in it are pretty much "perpendicular" to each other. I say that in quotes, because that is what orthogonal means. The principle has just been generalized to various sets of different dimensions instead of just two dimensions.