Gram-Schmidt Calculator
Perform Gram-Schmidt orthogonalization on vectors with step-by-step solutions. Get orthogonal and orthonormal bases.
Gram-Schmidt Calculator
Perform Gram-Schmidt orthogonalization on vectors with step-by-step solutions
Vector Input
Vector Settings
Dimension:
3
Number of Vectors:
3
Vector Components
v1:
v2:
v3:
Gram-Schmidt Process Reference
| Step | Operation | Formula |
|---|---|---|
| 1 | First orthogonal vector | u₁ = v₁ |
| 2 | Projection | proj_u(v) = (v·u)/(u·u) × u |
| 3 | Subtract projection | u = v - proj_u(v) |
| 4 | Normalization | e = u / ||u|| |
Results
Your Gram-Schmidt results will appear here
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Step-by-step Gram-Schmidt orthogonalization process
Enter Your Vectors
Input your vectors with any number of components
Perform Orthogonalization
The calculator applies the Gram-Schmidt process
Get Orthogonal Basis
Receive orthogonal and orthonormal vectors with steps
Key Features
What makes our Gram-Schmidt Calculator special!
Multiple Dimensions
Works with vectors from 2D up to 5D
Step-by-Step Solutions
Detailed breakdown of each orthogonalization step
Visual Results
Clear display of orthogonal and orthonormal vectors
Who Can Benefit
Our Gram-Schmidt Calculator helps various users:
Students
Learn linear algebra concepts with concrete examples
Teachers
Demonstrate orthogonalization in classroom settings
Engineers
Create orthogonal bases for signal processing
Researchers
Verify manual calculations in academic work
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Try NowUser Reviews
Rated Excellent 4.98/5 on There's An AI For That (TAAFT) - The #1 AI Aggregator
- Jun 17, 2025
Saved My Linear Algebra Grade!
This calculator helped me understand Gram-Schmidt process for my exam.
Emily R.Verified Review - Jun 15, 2025
Professor Approved
I recommend this tool to all my linear algebra students.
Dr. Johnson, MathematicsVerified Review - Jun 13, 2025
Perfect for Homework
The step-by-step solutions show exactly how to do each calculation.
Daniel T.Verified Review - Jun 11, 2025
Great Learning Tool
Helped me visualize the orthogonalization process.
Sophia L.Verified Review
Frequently Asked Questions
Is this Gram-Schmidt Calculator free?
Yes, our Gram-Schmidt Calculator is completely free to use.
What is the Gram-Schmidt process?
The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space, commonly used to create orthogonal bases in linear algebra.
How many vectors can I input?
You can input between 2 and 5 vectors, each with 2 to 5 dimensions.
Does this show the calculation steps?
Yes, the calculator provides detailed step-by-step solutions showing each projection and orthogonalization step.
What's the difference between orthogonal and orthonormal?
Orthogonal vectors are perpendicular to each other, while orthonormal vectors are both orthogonal and normalized (have unit length).
Can I use this for complex vectors?
Currently this calculator works with real-valued vectors only. Complex vector support may be added in the future.
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