Find the orthonormal basis

[La_Lune]

Homework Statement

Consider R3 together with the standard inner product. Let A =
1 1 −1
2 1 3
1 2 −6

(a) Use the Gram-Schmidt process to find an orthonormal basis S1 for null(A), and an orthonormal basis
S2 for col(A).

(b) Note that S = S1 ∪ S2 is a basis for R3. Use the the Gram-Schmidt process to transform S into an orthonormal basis T for R3

Homework Equations

Wi=(1/||Vi||)Vi

The Attempt at a Solution

I think I know how to do the first part, but for part b the computation seems to get a bit messy so I doubt I might got the first part wrong...So after finding out the basis for S, do you just follow the regular process of Gram-Schmidt to find the orthonormal basis for R3?

Thanks!

 

[mighty2000]

For part b, you are correct in using the Gram-Schmidt process to transform the basis S into an orthonormal basis for R3. However, instead of using the regular process, you can use a modified version of the Gram-Schmidt process specifically for creating an orthonormal basis.

To do this, follow these steps:

1. Start with the first vector in S, let's call it v1. Normalize it by dividing it by its length ||v1|| to get w1.

2. For the next vector in S, v2, subtract the projection of v2 onto w1 from v2. This will give you a vector perpendicular to w1. Normalize this vector to get w2.

3. For the third vector in S, v3, subtract the projection of v3 onto both w1 and w2 from v3. This will give you a vector perpendicular to both w1 and w2. Normalize this vector to get w3.

4. Your orthonormal basis T will be {w1, w2, w3}.

This modified process will ensure that each vector in T is perpendicular to all the previous vectors, creating an orthonormal basis for R3. I hope this helps! Let me know if you have any further questions.