What is Gram-Schmidt Process?

The Gram-Schmidt process is a mathematical method used in linear algebra to convert a set of vectors into an orthogonal or orthonormal basis. This process is fundamental in various mathematical applications, including QR decomposition, least squares problems, and quantum mechanics.

The process works by taking a set of linearly independent vectors and creating a new set where each vector is orthogonal (perpendicular) to all previous vectors. The resulting vectors can then be normalized to create an orthonormal basis, where all vectors are both orthogonal and have a magnitude of 1.

How to Use the Gram-Schmidt Calculator

Step 1: Enter the number of vectors you want to orthogonalize in the “Number of Vectors” field.

Step 2: Input your vector components in the generated input fields. Each vector should be entered as comma-separated values.

Step 3: Select whether you want to generate an orthogonal basis or an orthonormal basis using the radio buttons.

Step 4: Click the “Calculate” button to process your vectors.

Step 5: Review the results displayed below, including:

• The original input vectors
• Step-by-step calculations showing the orthogonalization process
• The final orthogonal or orthonormal basis
• Verification of orthogonality through dot products

Step 6: Use the “Reset” button to clear all fields and start a new calculation.