Gram Schmidt Calculator

Gram Schmidt Calculator:

Use this Gram-Schmidt calculator to orthonormalize the set of vectors step by step and find the orthonormal basis in an inner product space.

How To Use The Gram-Schmidt Calculator?

Follow these steps:

Set the vector Size
Enter the components of each vector
Click on the “Calculate” button and get the orthonormal set of vectors with Gram Schmidt method

What Is The Gram-Schmidt Process?

In linear algebra and numerical analysis, the Gram-Schmidt process transforms a set of independent vectors into an orthonormal basis. This basis spans the same subspace as the original vector set. The process constructs an orthogonal set of vectors, meaning their dot product is zero. In addition to that, each vector in the new basis has a unit length (magnitude of 1). This orthonormal basis is helpful for certain applications of algebra and beyond. It is widely used in various fields like computer science, physics, statistics, and machine learning, as well as in QR decomposition, etc.

Gram-Schmidt Process Formula:

The Gram-Schmidt formula helps to find the projection of one vector onto another.

\(\ u_i = v_i - \sum_{j=1}^{i-1} \frac{v_i \cdot u_j}{\|u_j\|^2} u_j \)

What Does Orthogonal Mean?

In algebra, orthogonal means the relationship between two vectors that are perpendicular to each other. Two vectors are orthogonal if the result of their dot product is zero.

For instance, if you have two following vectors:

\(\vec{u_1}, \vec{v_1}\) Then if \(\vec{u_1} \dot\ \vec{v_1}=0\)

They will be considered orthogonal. When it comes to orthogonal basis visualization, they can be easily visualized in two-dimensional Euclidean spaces. However, you can apply it to vectors having any number of dimensions.

For the sets of vectors having higher or comple dimensions, determining the orthogonal basis can be difficult and time taking. In such cases, tools like an orthogonal basis calculator, which uses the Gram-Schmidt process, are beneficial. The Gram-Schmidt process is an algorithm that transforms a set of linearly independent vectors into an orthogonal (or orthonormal) basis for the span of the vectors.

What Is An Orthonormal Basis?

An orthonormal basis is a set of vectors within an inner product space (such as Euclidean space with the dot product) that spans the same space as the original vectors. These basis vectors are characterized by two key properties:

Orthogonality: Each vector is perpendicular to every other vector in the set
Normality: All the vectors have a unit length (norm of 1)

Using an orthonormal basis simplifies many vector space operations, including:

Vector Decomposition
Vector Projections
Solving Systems of Linear Equations

To facilitate the creation of such a basis, you can use an orthonormal basis calculator. It makes the calculations straightforward because the basis vectors are perpendicular to each other and have a unit length. This way, it forms a well-defined coordinate system for a vector space.

What Are The Steps Involved In Performing The Gram-Schmidt Process?

Start: Begin with the given set of linearly independent vectors \(\ {\vec{v_1},\ \vec{v_2},\ \vec{v_3},\ ...\vec{v_n}}\)
First Vector: The first orthogonal vector, \(\vec{u_1}\), is simply the first input vector: \(\vec{u_1} = \vec{v_1}\)
Subsequent Vectors (Orthogonalization): For each subsequent vector \(\vec{v_i}\) (where i > 1), we need to make it orthogonal to all the previously found orthogonal vectors \(\vec{u_j}\) (where j < i). We do this by subtracting the projections of \(\vec{v_i}\) onto each \(\vec{u_j}\)
The projection of \(\vec{v_i}\) onto \(\vec{u_j}\) is calculated using: \(\text{proj}_{\vec{u}_j} (\vec{v}_i) = \frac{\vec{v}_i \cdot \vec{u}_j}{\vec{u}_j \cdot \vec{u}_j} \vec{u}_j\)
The new orthogonal vector \(\vec{u_i}\) is then: \(\vec{u_i}= \vec{v_i}-\sum_{j=1}^{\ i-1} \text{proj}_{\vec{u}_j} (\vec{v}i)\) or \(\vec{u_i}= \vec{v_i}-\sum{j=1}^{\ i-1} \frac{\vec{v}_i \cdot \vec{u}_j}{|u_j|^2} u_j\)
Repeat: Continue this process for all remaining vectors in the input set
Normalization: To get the orthonormal basis, normalize each orthogonal vector \(\vec{u_i}\) by dividing it by its magnitude \(\ (| \vec{u_i}|): (\ \vec{e_i} = \frac{\vec{u_i}}{|\vec{u_i}|})\)
Result: The resulting set of vectors \(\ {\vec{e_1},\ \vec{e_2},\ \vec{e_3},\ ...\vec{e_n}}\) is the orthonormal basis

Example:

Orthonormalize the set of the vectors\(\ V_1 = \begin{bmatrix} 1 \\ 1 \\ \end{bmatrix} \ , V_2 = \begin{bmatrix} 2 \\ 1 \\ \end{bmatrix} \) , using the Gram-Schmidt process.

Solution:

Gram-Schmidt process,\(\vec{u_k} = \vec{v_k} - \sum_{j=1}^{k-1} \text{proj}_{\vec{u_j}} (\vec{v_k})\)

Where,

\(\ proj_{\vec{u_j}} (\vec{v_k}) = \frac{\vec{u_j} \cdot \vec{v_k}}{|\vec{u_j}|^2} \vec{u_j}\) is a vector projection.

The Normalized Vector is

\(\vec{e_k} = \frac{ \vec{u_k}}{|{\vec{u_k}}|}\)

Step 1:

\(\vec{u_1} = \vec{v_1} = \begin{bmatrix} 1\\ \ 1 \ \end{bmatrix}\)

\(\vec{e_1} = \frac{\vec{u_1}}{|\vec{u_1}|} = \frac{\begin{bmatrix} 1 \ 1 \end{bmatrix}}{\sqrt{1^2 + 1^2}} = \begin{bmatrix} \frac{1}{\sqrt{2}} \ \\\frac{1}{\sqrt{2}} \end{bmatrix} \approx \begin{bmatrix} 0.71 \\ 0.71 \\ \end{bmatrix}\)

Step #2: (Find Vector Projection)

\(\text{proj}_{\vec{u_1}} (\vec{v_2}) = \begin{bmatrix} 1.5 \\ 1.5 \\ \end{bmatrix}\)

Vector Subtraction:

\(\vec{u_2} = \vec{v_2} - \text{proj}_{\vec{u_1}} (\vec{v_2}) = \begin{bmatrix} 0.5 \\ -0.5 \end{bmatrix}\)

Calculate Unit Vector:

\(\vec{e_2} = \frac{ \vec{u_2}}{|{\vec{u_2}}|} \ = \ \begin{bmatrix} 0.71 \\ -0.71 \\ \end{bmatrix} \)

Result:

\(\begin{bmatrix} 0.71 \\ 0.71 \\ \end{bmatrix} \begin{bmatrix} 0.71 \\ -0.71 \\ \end{bmatrix} \)

Although the Gram-Schmidt process can be performed manually, specialized tools like a Gram Schmidt calculator can automate these calculations, especially for larger or more complex sets of vectors. They not only save time but also reduce the risk of human error.

Applications of The Gram-Schmidt Process:

The Gram-Schmidt process generates the orthonormal bases, and its applications are not limited to linear algebra. Let's take a look at some important areas where it plays a significant role:

1. Computer Graphics and 3D Modeling:

Generating Orthogonal Camera Frames: When dealing with 3D graphics, the orientation of the camera is settled by three orthogonal vectors. These perpendicular vectors prevent the rolling of the camera and help in providing clear, rendered images
Creating Surface Normals: In 3D rendering, orthogonal vectors play an important role in calculating the required lighting and shading

2. Signal Processing and Data Compression:

Wavelet Analysis: The Gram-Schmidt process creates orthogonal wavelet bases that compress and represent data more efficiently.
Filter Design: It is used to generate orthogonal filters in digital signal processing because of their properties

3. Quantum Mechanics:

Orthonormal Quantum States: The Gram-Schmidt process constructs orthonormal bases to simplify calculations and analysis of quantum systems
Atomic and Molecular Physics: It helps in creating orthonormal wave functions for describing electron states in atoms and molecules

4. Numerical Analysis and Linear Algebra:

QR Decomposition: The Gram-Schmidt process is very helpful in the QR decomposition of a matrix. This way, it helps to solve eigenvalue calculations and systems of linear equations
Least Squares Approximations: Gram-Schmidt makes it easy to find the line or curve that best matches data

5. Machine Learning and Data Science:

Feature Orthogonalization: The use of orthogonal in machine learning helps in improving the performance, condition of data, and eliminating the redundancy

Principal Component Analysis (PCA) Related Tasks: The Gram-Schmidt process is related to the underlying math behind PCA

Why Choose Our Gram Schmidt Calculator?

Step-By-Step Breakdown: Understand every step of calculation
Fast and Efficient: Get instant results
Accurate Results: Reliable and precise orthonormal bases
User-Friendly Interface: Easy input for users of all levels
Completely Free: No cost for full functionality
24/7 Access: Use it anytime, anywhere

FAQ’s:

How To Find Orthonormal Basis Using Gram-Schmidt Process?

1. Start with a set of linearly independent vectors:

\(\vec{u}_1,\ \vec{u}_2,\ \vec{u}_3,\ \ldots,\ \vec{u}_n\)

2. Set:

\(\vec{v}_1 = \vec{u}_1\)

Then normalize:

\(\vec{e}_1 = \frac{\vec{v}_1}{\|\vec{v}_1\|}\)

3. Compute the orthogonal vector for** \( \vec{u}_2 \):

\(\vec{v}_2 = \vec{u}_2 - \mathrm{proj}_{\vec{v}_1} \vec{u}_2 = \vec{u}_2 - \frac{\vec{u}_2 \cdot \vec{v}_1}{\vec{v}_1 \cdot \vec{v}_1} \vec{v}_1\)

Normalize it:

\(\vec{e}_2 = \frac{\vec{v}_2}{\|\vec{v}_2\|}\)

4. For \( \vec{u}_3 \), subtract the projections onto both \( \vec{v}_1 \) and \( \vec{v}_2 \):

\(\vec{v}_3 = \vec{u}_3 - \mathrm{proj}_{\vec{v}_1} \vec{u}_3 - \mathrm{proj}_{\vec{v}_2} \vec{u}_3\)

Normalize it:

\(\vec{e}_3 = \frac{\vec{v}_3}{\|\vec{v}_3\|}\)

5. For each \( \vec{u}_i \), subtracting projections onto all previous \( \vec{v}_j \) (for \( j < i \)), and normalize:

\(\vec{v}_i = \vec{u}_i - \sum_{j=1}^{i-1} \mathrm{proj}_{\vec{v}_j} \vec{u}_i = \vec{u}_i - \sum_{j=1}^{i-1} \left( \frac{\vec{u}_i \cdot \vec{v}_j}{\vec{v}_j \cdot \vec{v}_j} \vec{v}_j \right)\)

The following set forms the orthonormal basis:

\(\vec{e}_1, \vec{e}_2, \ldots, \vec{e}_n\)

To streamline this process, especially with numerous or high-dimensional vectors, try our Gram Schmidt calculator. It instantly delivers precise orthonormal bases with clear, step-by-step explanations.

Can I Apply Gram-Schmidt To Linearly Dependent Vectors?

Yes, you can apply the Gram-Schmidt Process to linearly dependent vectors but it will result in a zero vector during the process, making normalization impossible. In simple words, it finds linear dependency, but it can't produce the full basis if the vectors are not linearly dependent.

What Is The Main Difference Between The Orthogonal And Orthonormal?

Orthogonal vectors are 90 degrees from each other. Orthonormal vectors are also orthogonal and have a vector length of 1.

Does Order Matter For Gram-Schmidt?

The order of the vectors does not affect the resulting subspace spanned by the orthonormal basis. However, it can affect the decomposition pattern of vectors you end up with on that basis.

What Is The Function of Gram Schmidt Calculator?

A Gram-Schmidt calculator implements the Gram-Schmidt orthogonalization process on a set of linearly independent vectors. Its primary function is to:

Accept Input: A set of linearly independent vectors or the columns of a matrix
Apply the Gram-Schmidt Process: Generate an orthonormal basis for the vector space spanned by the input vectors
Provide Output: The resulting orthonormal basis vectors

References:

From the source of Wikipedia: Gram–Schmidt process

From the source of math.hmc.edu: Gram–Schmidt Method