Gram Schmidt Calculator

Introduction to Gram Schmidt Calculator With Steps:

Gram schmidt calculator is an online tool that helps you to find the orthonormal basis from a given set of vectors in a few seconds. Our tool uses the orthogonalization method on a linearly independent vector that is space to the entire set to get the solution.

The Gram-Schmidt process efficiently solves complex vector problems. If you try to evaluate vector problems manually, you will get stuck because you do not know the base of vector space due to the complicated method, so you will need to use the gram schmidt process calculator to get a solution.

What is the Gram-Schmidt Process?

The Gram-Schmidt method is a technique within vector spaces used to convert a set of linearly independent vectors into an orthogonal (or orthonormal) set of vectors in an inner product space that spans the given vector. It is named after mathematicians Pedersen Gram and Erhard Schmidt, who developed the process of solving inner products.

Gram-Schmidt process is a fundamental process because it is widely applied in fields like physics, engineering, and computer science.

$$ u_i \;=\; v_i - \sum_{j = 1}^{i - 1} \frac{〈v_i , u_j〉}{〈u_j , u_j〉} u_j $$

Here v1,v2,...,vn is the given vector in a vector space. U1,u2,..,un is the orthogonal vector that is supposed to check its linear independence.

What is the Gram-Schmidt Orthogonalization Procedure?

The Gram-Schmidt orthogonalization procedure transforms a set of linearly independent vectors into an orthogonal (or orthonormal) set of vectors in an inner product space.

Here is a procedure that helps you to understand this method with an example.

Example: Find the given vectors

$$ v_1 \;=\; \left[ \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} \right], v_2 \;=\; \left[ \begin{matrix} 1 \\ 1 \\ 0 \\ \end{matrix} \right] \;and\; v_3 \;=\; \left[ \begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix} \right] $$

Solution:

Given vector is v1 , v2 and v3

Let {u1,…,ui−1} is the orthogonal vector for each i=2,3,…,n that means v1=u1,....,vi=ui as,

$$ u_1 \;=\; \left[ \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} \right] $$

Compute ui when you subtract the projections of vi onto orthogonalized vectors [u1,…,ui−1] with the help of a given formula for the inner product.

$$ u_i \;=\; v_i - \sum_{j = 1}^{i - 1} \frac{〈v_i , u_j〉}{〈u_j , u_j〉}u_j $$

For this question our general formula of the inner product becomes,

$$ u_2 \;=\; v_2 - \frac{〈v_2 , u_1〉}{〈u_1 , u_1〉}u_1 $$

To find the u2 value add all the given values,

$$ 〈v_2 , u_1〉\;=\; 1 . 1 + 1 . 0 + 0 . 0 \;=\; 1 $$

$$ 〈u_1 , u_1〉 \;=\; 1 $$

$$ u_2 \;=\; \left[ \begin{matrix} 1 \\ 1 \\ 0 \\ \end{matrix} \right] - \frac{1}{1} \left[ \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix} \right] $$

To find the u3 value again add values in the formula,

$$ u_3 \;=\; v_3 - \frac{〈v_3 , u_1〉}{〈u_1 , u_1〉} u_1 - \frac{〈v_3 , u_2〉}{〈u_2 , u_2〉} u_2 $$

Calculate the inner product,

$$ 〈v_3 , u_1〉\;=\; 1 . 1 + 1 . 0 + 1 . 0 \;=\; 1 $$

$$ 〈v_3 , u_1〉\;=\; 1 . 0 + 1 . 1 + 1 . 0 \;=\; 1 $$

$$ 〈u_2 , u_2〉\;=\; 1 $$

We get,

$$ u_3 \;=\; \left[ \begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix} \right] - \frac{1}{1} \left[ \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} \right] - \frac{1}{1} \left[ \begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 0 \\ 0 \\ 1 \\ \end{matrix} \right] $$

Therefore the solution of orthogonal vectors to each other spans the entire vectors {v1,v2,v3}.

How to Find the Gram Schmidt Orthonormal Basis?

To find the gram schmidt orthonormal basis you first need to find the orthogonal vector. After that use an orthogonal vector to find the orthonormal basis.

Let's take an example where orthogonal vectors are already by using the above procedure. For example:

$$ v_1 \;=\; \left[ \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} \right], v_2 \;=\; \left[ \begin{matrix} 1 \\ 1 \\ 0 \\ \end{matrix} \right] \;and\; v_3 \;=\; \left[ \begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix} \right] $$

Solution:

The given vector orthogonal solution is,

$$ u_1 \;=\; \left[ \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} \right] $$

$$ u_2 \;=\; \left[ \begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix} \right] $$

$$ u_3 \;=\; \left[ \begin{matrix} 0 \\ 0 \\ 1 \\ \end{matrix} \right] $$

For the orthonormal vector, we need to find {e1,e2,e3}, first, we should know u1=e1 which means we just need to find e2 and e3 values. For that,

$$ e_2 \;=\; \frac{u_2}{|| u_2 ||} $$

$$ ‖ u_2‖ \;=\; √1 + 1 + 0 \;=\; √2 $$

Put value in the above formula to find e2.

$$ e_2 \;=\; \frac{u_2}{||u_2||} \;=\; \frac{1}{\sqrt{2}} \left[ \begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 0 \\ \frac{1}{\sqrt{2}} \\ 0 \\ \end{matrix} \right] $$

$$ e_3 \;=\; \frac{u_3}{||u_3||} $$

$$ ‖ u_3‖ \;=\; √0 + 1 + 0 \;=\; √1 \;=\; 1 $$

Put value in the above formula to find e3.

$$ e_3 \;=\; \frac{u_3}{||u_3||} \;=\; \left[ \begin{matrix} 0 \\ 0 \\ 1 \\ \end{matrix} \right] $$

Therefore, the orthonormal basis of {v1,v2,v3} using the gram-schmidt method is,

$$ \{e_1, e_2, e_3 \} \;=\; \left[ \left( \begin{matrix}1 \\ 0 \\ 0 \\ \end{matrix} \right), \left( \begin{matrix} 0 \\ \frac{1}{\sqrt{2}} \\ 0 \\ \end{matrix} \right), \left( \begin{matrix} 0 \\ 0 \\ 1 \\ \end{matrix} \right) \right] $$

Alternatively, you can use our gram schmidt calculator with steps to find the gram schmidt orthonormal basis in a few clicks for free.

How to Use the Gram-Schmidt Calculator?

Orthogonal basis calculator has a simple design that makes it easy for you to understand how to use it for evaluating the projection of a set of vectors only by following our instructions that are:

1. Enter the number of vectors in the input field.
2. Enter the size of the vector in the input field.
3. Enter the value of the vector to find the orthogonalization in the next input box.
4. Review the given vector before hitting the calculate button to start the evaluation process in the orthonormal basis calculator.
5. Click the “Calculate” button to get the result of your given vector problem that spans the vector v1,v2,..vi.
6. If you are trying our gram-schmidt process calculator for the first time then you can use the load example to learn more about this method.
7. Click on the "Recalculate" button to get a new page for finding more example solutions of gram schmidt problems from the given vector.

Output From Orthogonal Matrix Calculator:

Orthonormal matrix calculator gives you the solution from a given vector when you add the input into it. It provides you with solutions that is:

• Result Option:

When you click on the result option the gram schmidt calculator gives you a solution to the projection of the vector problem.

• Possible Steps:

When you click on it, this option will provide you with a solution where all the calculations of gram schmidt process steps are mentioned.

Key Features of Using Gram Schmidt Process Calculator:

The gram-schmidt calculator provides you with many useful features that help you to calculate vector problems and give you a solution without any trouble. These features are:

• Orthogonal basis calculator is a free-of-cost tool so you can use it for free to find orthonormal vector problem solutions without paying.
• It is an adaptable tool that can manage various types of vectors to calculate the orthogonalization of a vector space.
• Our orthonormal basis calculator helps you to get conceptual clarity for the Gram Schmidt process when you use it for practice by solving more examples.
• It saves the time that you consume on the calculation of the inner product of vector problems.
• The gram-schmidt process calculator is a reliable tool that provides you with accurate solutions whenever you use it to calculate the orthonormal vectors without any man-made errors in calculation.
• Orthogonal matrix calculator provides the solution without imposing any striction which means you can use it multiple times.
• Our tool ensures efficiency and accuracy in solving vector-related tasks using the gram-schmidt method.