Solving Orthonormal System with Gram-Schmidt

  • Thread starter Thread starter Niles
  • Start date Start date
  • Tags Tags
    System
Click For Summary
SUMMARY

The discussion centers on using the Gram-Schmidt process to convert a set of vectors, specifically u_1, u_2, and u_3 in R^4, into an orthonormal system v_1, v_2, and v_3. The user, Niles, correctly identifies that applying the Gram-Schmidt algorithm will yield the desired orthonormal vectors while maintaining the same span. This method is confirmed as the appropriate approach to achieve the goal of creating an orthonormal basis from the given vectors.

PREREQUISITES
  • Understanding of vector spaces and spans
  • Familiarity with the Gram-Schmidt process
  • Knowledge of orthonormal vectors
  • Basic linear algebra concepts
NEXT STEPS
  • Study the detailed steps of the Gram-Schmidt algorithm
  • Explore applications of orthonormal systems in linear transformations
  • Learn about the geometric interpretation of orthonormal bases
  • Investigate the implications of orthonormality in numerical methods
USEFUL FOR

Students and educators in linear algebra, mathematicians, and anyone seeking to understand the process of orthonormalization using Gram-Schmidt.

Niles
Messages
1,834
Reaction score
0
[SOLVED] Orthonormal system

Homework Statement


I have three vectors, u_1, u_2 and u_3, in R^4. I want to find an orthonormal system v_1, v_2 and v_3, where:

span(u_1, u_2, u_3) = span(v_1, v_2, v_3).

The Attempt at a Solution


My "guess" is that I can just use Gram-Schmidt on u_1, u_2 and u_3 to find the orthonormal system, and the two vector-sets must span the same space, right?

Thanks in advance,

sincerely Niles.
 
Physics news on Phys.org
Yes, you are supposed to use Gram-Schmidt.
 

Similar threads

Minimize integral using orthogonal basis
  • · Replies 2 ·
Replies
2
Views
1K
Partial differential equations, symmetries, invariants, conservations,
  • · Replies 2 ·
Replies
2
Views
1K
Basis and Components (Vectors)
  • · Replies 5 ·
Replies
5
Views
2K
Prove :Union of Three subspaces is a subspace if ....
  • · Replies 3 ·
Replies
3
Views
4K
Dimension of orthogonal subspaces sum
  • · Replies 4 ·
Replies
4
Views
2K
Probability theory: Understanding some steps
  • · Replies 6 ·
Replies
6
Views
2K
I cannot make the Gram-Schmidt procedure work
  • · Replies 16 ·
Replies
16
Views
3K
Linear Algebra - Gram-Schmidt Process
  • · Replies 4 ·
Replies
4
Views
2K
Linear Combination Proof of Orthonormal basis
  • · Replies 11 ·
Replies
11
Views
3K
Gram-Schmidt for 1, x, x^2 Must find orthonormal basis
  • · Replies 5 ·
Replies
5
Views
6K