📐 Gram-Schmidt Calculator
Orthogonalize vectors using the Gram-Schmidt process
How to Use This Calculator
Select Dimensions
Choose vector dimension (2D, 3D, or 4D) and number of vectors.
Enter Vectors
Input components of each vector v₁, v₂, ..., vₙ.
Calculate
Click to perform Gram-Schmidt process and get orthogonal/orthonormal vectors.
Review Results
Orthogonal vectors (uᵢ) are perpendicular. Orthonormal vectors (eᵢ) are also normalized to length 1.
Formula
u₁ = v₁
uᵢ = vᵢ - Σⱼ₌₁ⁱ⁻¹ proj{uⱼ}(vᵢ)
eᵢ = uᵢ / ||uᵢ|| (orthonormal)
Gram-Schmidt Process:
- Set u₁ = v₁
- For i > 1: uᵢ = vᵢ - Σⱼ₌₁ⁱ⁻¹ (⟨vᵢ, uⱼ⟩ / ⟨uⱼ, uⱼ⟩) uⱼ
- For orthonormal: eᵢ = uᵢ / ||uᵢ||
Projection Formula:
proj_u(v) = (⟨v, u⟩ / ⟨u, u⟩) u
where ⟨v, u⟩ is the dot product
Properties:
- uᵢ ⟂ uⱼ for i ≠ j (orthogonal)
- ||eᵢ|| = 1 for all i (orthonormal)
- span{u₁, ..., uₖ} = span{v₁, ..., vₖ}
About Gram-Schmidt Calculator
The Gram-Schmidt Calculator orthogonalizes a set of vectors using the Gram-Schmidt process. Given linearly independent vectors v₁, v₂, ..., vₙ, it produces orthogonal vectors u₁, u₂, ..., uₙ that span the same space. Normalizing these gives orthonormal vectors.
When to Use This Calculator
- QR Decomposition: Orthogonalize columns for QR factorization
- Orthogonal Bases: Create orthogonal basis from given vectors
- Projections: Find orthogonal projections
- Linear Algebra: Understand orthogonalization process
- Signal Processing: Orthogonalize signal components
Why Use Our Calculator?
- ✅ Complete Process: Shows orthogonal and orthonormal vectors
- ✅ Multiple Dimensions: Supports 2D, 3D, and 4D vectors
- ✅ Step Display: Shows projection steps
- ✅ Educational: Helps understand Gram-Schmidt algorithm
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Orthogonal: Vectors are perpendicular: uᵢ · uⱼ = 0 for i ≠ j
- Orthonormal: Orthogonal + unit length: ||eᵢ|| = 1
- Projection: Component of one vector along another
- Span Preservation: Gram-Schmidt preserves the span of original vectors
- Linearly Independent: Required for successful orthogonalization
Applications
QR Decomposition: A = QR where Q is orthonormal (from Gram-Schmidt) and R is upper triangular.
Least Squares: Orthogonal bases simplify least squares problems.
Frequently Asked Questions
What is the Gram-Schmidt process?
Gram-Schmidt is an algorithm that converts a set of linearly independent vectors into an orthogonal (or orthonormal) set spanning the same space. It works by subtracting projections onto previous vectors.
What's the difference between orthogonal and orthonormal?
Orthogonal vectors are perpendicular (dot product = 0). Orthonormal vectors are orthogonal AND have unit length (norm = 1). Orthonormal = orthogonal + normalized.
What if vectors are linearly dependent?
If vectors are linearly dependent, one of the orthogonal vectors uᵢ will be (or nearly) zero. The process still works but produces fewer non-zero orthogonal vectors than input vectors.
Why use Gram-Schmidt?
Orthogonal bases simplify many computations: dot products become simpler, projections are easier, and numerical stability is better. It's essential for QR decomposition.
Is the result unique?
The orthogonal basis is not unique - it depends on the order of input vectors. However, the span is preserved, and all valid orthogonal bases span the same space.