Gram-Schmidt Calculator

Orthogonalize two 2D vectors using the Gram-Schmidt process.

Linear AlgebraVectorsCollege Math

Calculator

v₁ x
v₁ y
v₂ x
v₂ y
Please enter valid values.

What this calculator teaches

Gram-Schmidt turns independent vectors into orthogonal vectors that span the same space.

It is the core process behind QR decomposition and many numerical algorithms.

Formula & Symbols

ConceptFormulaMeaning
Projection removalu₂=v₂-proj_u₁(v₂)Removes the component of v₂ in the u₁ direction.
Normalizationq=u/||u||Makes an orthogonal vector unit length.

Step-by-Step Examples

Example 1
v₁=(1,1), v₂=(1,0)
  • u₁=(1,1).
  • Subtract projection of v₂ onto u₁.
  • Normalize both vectors.
✓ The calculator reports orthogonal and orthonormal vectors.

Where students use this

🎓
College mathematics
Use the tool to connect notation, formulas, and numeric answers.
🧪
Engineering analysis
Apply the result to models, systems, approximations, and design calculations.
💻
Computer science
Use matrix, vector, numerical, and optimization ideas in algorithms and data work.

Common Mistakes to Avoid

⚠️
Ignoring restrictions

Check denominator, domain, endpoint, and matrix-invertibility conditions before trusting a result.

⚠️
Rounding too early

Carry extra decimals through intermediate steps and round only the final answer.

⚠️
Mixing symbolic and numeric answers

Some topics need exact symbolic reasoning, while this calculator gives a clear numerical learning check.

Frequently Asked Questions

Is this calculator meant for students? ▾
Yes. It gives an answer and shows the main method so students can learn the reasoning instead of copying only the result.
Can I use decimals and negative numbers? ▾
Yes, when the topic allows them. Some formulas have restrictions such as positive radii, nonzero denominators, or valid intervals.
Why can textbook answers look different? ▾
Equivalent algebraic forms, different rounding, or using exact notation instead of decimals can make answers look different while meaning the same thing.

Related Math Calculators

🔢 Vector Projection Calculator🔢 QR Decomposition Calculator🔢 Dot Product Calculator

Formula Explorer connections

Interpretation: This relationship represents vectors, linear systems or transformations using matrix structure and vector operations. Assumption: Matrix dimensions must be compatible. Inversion, decomposition and unique solutions require rank, conditioning or nonzero-determinant conditions.

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