Orthogonal Basis Calculator – Gram-Schmidt Process
Calculate orthogonal and orthonormal bases from vectors
Table of Contents
How to Use
- Enter the x, y, and z components of the first vector
- Enter the x, y, and z components of the second vector
- Enter the x, y, and z components of the third vector
- Click calculate to see the orthogonal and orthonormal bases
What is an Orthogonal Basis?
An orthogonal basis is a set of vectors that are mutually perpendicular (orthogonal) to each other. When these vectors also have unit length (magnitude of 1), they form an orthonormal basis. These concepts are fundamental in linear algebra and have wide applications in mathematics, physics, and engineering.
The Gram-Schmidt process is an algorithm that takes a set of linearly independent vectors and produces an orthogonal (or orthonormal) set of vectors that span the same subspace.
The Gram-Schmidt Process
Given vectors v₁, v₂, v₃, the Gram-Schmidt process constructs orthogonal vectors u₁, u₂, u₃ as follows:
- u₁ = v₁ (first vector remains unchanged)
- u₂ = v₂ - proj(v₂, u₁) (subtract projection onto u₁)
- u₃ = v₃ - proj(v₃, u₁) - proj(v₃, u₂) (subtract projections onto u₁ and u₂)
To obtain an orthonormal basis, each vector is normalized by dividing by its magnitude: eᵢ = uᵢ / ||uᵢ||
Applications of Orthogonal Bases
- QR Decomposition: Used in solving linear systems and eigenvalue problems
- Computer Graphics: Coordinate transformations and camera systems
- Signal Processing: Fourier analysis and wavelet transforms
- Machine Learning: Principal Component Analysis (PCA)
- Quantum Mechanics: State vectors and measurement bases
- Numerical Analysis: Least squares approximations
Properties of Orthogonal Bases
- Perpendicularity: All pairs of basis vectors have zero dot product
- Linear Independence: Orthogonal vectors are always linearly independent
- Easy Projections: Projecting onto orthogonal bases is computationally simple
- Coordinate Computation: Finding coordinates is straightforward using dot products
- Stability: Orthonormal bases provide numerical stability in computations
Frequently Asked Questions
- What is the difference between orthogonal and orthonormal bases?
- An orthogonal basis consists of mutually perpendicular vectors of any length. An orthonormal basis is an orthogonal basis where each vector has been normalized to have unit length (magnitude of 1). Both are useful, but orthonormal bases simplify many calculations.
- Why is the Gram-Schmidt process important?
- The Gram-Schmidt process is fundamental because it provides a systematic way to construct orthogonal bases from any set of linearly independent vectors. This is essential for QR decomposition, solving least squares problems, and many other applications in numerical linear algebra.
- What happens if the input vectors are linearly dependent?
- If the input vectors are linearly dependent, the Gram-Schmidt process will produce a zero vector at some step. This indicates that the vectors do not span a full 3D space and cannot form a complete basis for R³.
- Can this process be extended to more than 3 dimensions?
- Yes, the Gram-Schmidt process works in any number of dimensions. The algorithm remains the same: each new vector has the projections onto all previous orthogonal vectors subtracted from it.