Gram-Schmidt Orthonormalization Calculator

Convert a set of linearly independent vectors into an orthogonal or orthonormal basis.

Enter your vectors below, one vector per line. Numbers in a vector should be separated by commas or spaces. The calculator will apply the Gram-Schmidt process to generate two sets of basis vectors.

The number of dimensions will be inferred from the first vector.

Examples

Click on any example to load it into the calculator.

Basic 2D Vectors

2d-vectors

A simple case with two vectors in a 2D space.

Vectors:\n3, 1
2, 2

Standard 3D Basis

3d-vectors

Applying the process to three vectors in 3D.

Vectors:\n1, 1, 1
1, 0, 1
-1, 1, 0

Four 4D Vectors

4d-vectors

A more complex example in a 4-dimensional space.

Vectors:\n1, 0, 1, 0
1, 1, 1, 1
0, 1, 2, 1

Linearly Dependent Set

dependent-vectors

An example with linearly dependent vectors to show how the process handles it.

Vectors:\n1, 1, 0
2, 2, 0
1, 0, 1
Other Titles
Understanding Gram-Schmidt Orthonormalization: A Comprehensive Guide
A deep dive into the Gram-Schmidt process, its mathematical foundations, applications, and step-by-step calculations.

What is the Gram-Schmidt Process?

  • A method for converting a set of vectors into an orthogonal or orthonormal set.
  • A cornerstone of linear algebra for creating simplified vector bases.
  • Transforms a basis into a more structured and computationally useful form.
The Gram-Schmidt process is an algorithm used in linear algebra to 'clean up' a set of vectors. Given a finite, linearly independent set of vectors in an inner product space, it generates an orthogonal set (where all vectors are at right angles to each other) and an orthonormal set (where the orthogonal vectors are also unit vectors, i.e., have a length of 1).
This transformation is immensely useful because working with orthogonal or orthonormal bases simplifies many mathematical computations, from solving systems of linear equations to performing data analysis with techniques like QR decomposition.
The Core Idea: Vector Projection
The process works by iteratively taking a vector and subtracting its projections onto the previously processed vectors. This removes any component of the current vector that lies in the direction of the previous ones, ensuring the new vector is orthogonal to all of them.

Fundamental Concepts

  • Basis {v1, v2} becomes {u1, u2} where u1 · u2 = 0.
  • If v1 = (3, 1), v2 = (2, 2), the process yields orthogonal vectors like u1 = (3, 1) and u2 = (-0.4, 1.2).
  • Normalizing the orthogonal vectors gives vectors of length 1.

Step-by-Step Guide to Using the Gram-Schmidt Calculator

  • Learn the correct input format for your vectors.
  • Understand how to interpret the orthogonal and orthonormal results.
  • Troubleshoot common issues like linear dependency.
Our calculator simplifies the Gram-Schmidt process into a few easy steps, allowing you to focus on the results rather than the manual computation.
Input Guidelines:
  • Vector Entry: Enter one vector per line in the text area. For example, to input three 3D vectors, you would have three lines of text.
  • Number Format: Separate the numbers (components) within each vector using either commas (,) or spaces. For instance, '1, 2, 3' and '1 2 3' are both valid.
  • Dimensionality: Ensure all vectors have the same number of components. The calculator determines the dimension from the first vector you enter.
Interpreting the Results:
  • Orthogonal Basis: This first set of results contains vectors that are mutually perpendicular (their dot product is zero). These vectors are not normalized.
  • Orthonormal Basis: This second set contains vectors that are not only perpendicular but also have a length of 1. This is often the desired result for most applications.
  • Linear Dependency Warning: If the input vectors are linearly dependent, the process will produce at least one zero vector. Our calculator will flag this with a warning.

Practical Usage Examples

  • Input: 1,1 1,0 Output (Orthonormal): [0.707, 0.707] [0.707, -0.707]
  • Input: 1,0,0 1,1,0 1,1,1 Output (Orthogonal): [1,0,0] [0,1,0] [0,0,1]
  • Inputting linearly dependent vectors like '1,1' and '2,2' will result in a zero vector.

Real-World Applications of Gram-Schmidt

  • Computer graphics and 3D modeling.
  • Machine learning and data science.
  • Signal processing and communications.
The Gram-Schmidt process is not just an abstract mathematical tool; it's a workhorse in many fields of science and engineering.
QR Decomposition
One of the most important applications is in QR decomposition, where a matrix A is factored into the product of an orthogonal matrix Q and an upper triangular matrix R. The columns of Q are the orthonormal vectors obtained by applying the Gram-Schmidt process to the columns of A. This decomposition is widely used for solving linear systems and eigenvalue problems.
Machine Learning and Statistics
In Principal Component Analysis (PCA), a technique used for dimensionality reduction, orthogonal bases are essential for finding the directions of maximum variance in a dataset. Gram-Schmidt can be used to construct these bases.
Computer Graphics
In 3D graphics, creating a coordinate system or 'camera view' relative to an object requires an orthonormal basis (for up, right, and forward vectors). Gram-Schmidt is a perfect tool for creating this basis from potentially non-orthogonal starting vectors.

Industry Applications

  • Finding the closest point in a subspace to a given point.
  • Generating polynomial approximations of functions.
  • Creating orthogonal codes for CDMA in mobile communications.

Common Misconceptions and Correct Methods

  • Order of vectors matters significantly.
  • Numerical instability with nearly dependent vectors.
  • Distinguishing between orthogonal and orthonormal.
While powerful, the Gram-Schmidt process has nuances that can lead to confusion or incorrect results if not properly understood.
Misconception 1: The Order of Vectors Doesn't Matter
This is incorrect. Changing the order of the input vectors {v1, v2, ..., vk} will result in a different final orthogonal/orthonormal basis. While both bases will span the same subspace, the individual vectors will be different. The first vector in the output basis is always aligned with the first vector in the input.
Misconception 2: It Works for Any Set of Vectors
The standard Gram-Schmidt process requires the input vectors to be linearly independent. If they are dependent, at some point the algorithm will attempt to find a vector that is a linear combination of the previous ones, resulting in a zero vector. While our calculator handles this, it signifies an issue with the initial set.
Misconception 3: Orthogonal and Orthonormal are the Same
Orthogonal means all vectors are perpendicular (dot product is 0). Orthonormal means they are orthogonal AND each vector has a length of 1. The orthonormal basis is usually more useful but requires the extra step of normalization.
A common issue is numerical instability. When vectors are 'almost' linearly dependent, subtraction can lead to a loss of precision. More stable versions, like the Modified Gram-Schmidt process, are often used in high-precision software.

Clarification Examples

  • Processing {v1, v2} gives a different result than processing {v2, v1}.
  • Inputting {(1,0), (0,1), (1,1)} will produce a zero vector for the third output.
  • Orthogonal vector (2,0) becomes orthonormal vector (1,0) after normalization.

Mathematical Derivation and Formulas

  • The projection formula at the heart of the process.
  • The iterative algorithm for building the orthogonal basis.
  • The final normalization step.
The elegance of the Gram-Schmidt process lies in its straightforward iterative construction, which is built upon the concept of vector projection.
The Projection Formula
The projection of a vector v onto another vector u is given by the formula: proj_u(v) = ((v · u) / (u · u)) * u. This formula calculates the 'shadow' that vector v casts onto the line defined by vector u.
The Algorithm
Let the initial set of linearly independent vectors be {v1, v2, ..., vk}. We find the orthogonal basis {u1, u2, ..., uk} as follows:
1. u1 = v1
2. u2 = v2 - proj_u1(v2)
3. u3 = v3 - proju1(v3) - proju2(v3)
4. ...
5. uk = vk - Σ(from j=1 to k-1) proj_uj(vk)
At each step, we take the next original vector (vi) and subtract its projections onto all the orthogonal vectors we have already found (u1, ..., ui-1). The result is a new vector (ui) that is guaranteed to be orthogonal to all previous ones.
Normalization
To get the orthonormal basis {e1, e2, ..., ek}, we simply divide each orthogonal vector by its magnitude (norm): ei = ui / ||ui||, where ||ui|| = sqrt(ui1^2 + ui2^2 + ...).