What is Gram-Schmidt ortogonalization?

The Gram-Schmidt orthogonalization is a mathematical procedure that allows you to find the orthonormal basis of the vector space defined by a set of vectors. The orthonormal basis is a minimal set of vectors whose combinations span the entire space. Read more

How do I perform the Gram-Schmidt ortogonalization?

From a set of vectors v₁ , v₂ , and v₃ , we can find the elements of the orthonormal basis: Set u₁ = v₁ . Normalize u₁ : e₁ = u₁/|u₁| . This is the first element of the orthonormal basis. Find the second vector of the basis by choosing the vector orthogonal to u₁ : u₂ = v₂ - [(v₂ ⋅ u₁)/(u₁ ⋅ u₁)] × u₁ . Normalize u₂ : e₂ = u₂/|u₂| . Repeat the procedure for v₃ . Finding the null vector means that the original set is not linearly independent: they span a two-dimensional vector space. Read more

How do I find the second base vector if v₂=(4,2,1) and u₁=(3,-2,4)?

Starting from the first orthogonal vector u₁=(3,-2,4) , follow these steps: Calculate the projection of v₂=(4,2,1) onto u₁ : proj(v₂) = [(v₂ ⋅ u₁)/(u₁ ⋅ u₁)] × u₁ = 12/29 × (4,2,1) = (1.24,-0.83,1.66) Subtract the projection from v₂ : v₂ - proj(v₂) = (4,2,1) - (1.24,-0.83,1.66) = (2.76, 2.83, -0.66) The result is the vector u₂ . To confirm their orthogonality, you can compute the dot product and verify that it is equal to zero. Read more

Can I apply Gram-Schmidt to linearly dependent vectors?

Yes, but applying the Gram-Schmidt orthonormalization to linearly dependent vectors necessarily reduces the number of vectors in the original set. The orthonormalization procedure gives you the minimal set of vectors that can generate the original set: if there is redundancy in the latter, the process eliminates it. Read more

Gram-Schmidt Calculator

Created by Maciej Kowalski, PhD candidate

Reviewed by Bogna Szyk and Jack Bowater

Last updated: Mar 01, 2023

Table of contents:

Welcome to the Gram-Schmidt calculator, where you'll have the opportunity to learn all about the Gram-Schmidt orthogonalization. This simple algorithm is a way to read out the orthonormal basis of the space spanned by a bunch of random vectors. If you're not too sure what orthonormal means, don't worry! It's just an orthogonal basis whose elements are only one unit long. And what does orthogonal mean? Well, we'll cover that one soon enough!

So, just sit back comfortably at your desk, and let's venture into the world of orthogonal vectors!

What is a vector?

One of the first topics in physics classes at school is velocity. Once you learn the magical formula of $v = s / t$ , you open up the exercise book and start drawing cars or bikes with an arrow showing their direction parallel to the road. The teacher calls this arrow the velocity vector and interprets it more or less as "the car goes that way."

You can find similar drawings throughout all of physics, and the arrows always mean which direction a force acts on an object and how large it is. The scenario can describe anything from buoyancy in a swimming pool to the free fall of a bowling ball, but one thing stays the same: whatever the arrow is, we call it a vector.

In full (mathematical) generality, we define a vector to be an element of a vector space. In turn, we say that a vector space is a set of elements with two operations that satisfy some natural properties. Those elements can be quite funky, like sequences, functions, or permutations. Fortunately, for our purposes, regular numbers are funky enough.

Cartesian vector spaces

A Cartesian space is an example of a vector space. This means that a number, as we know them, is a (1-dimensional) vector space. The plane (anything we draw on a piece of paper), i.e., the space a pairs of numbers occupy, is a vector space as well. And lastly, so is []the 3-dimensional space of the world we live in, interpreted as a set of three real numbers.

When dealing with vector spaces, it's important to keep in mind the operations that come with the definition: addition and multiplication by a scalar (a real or complex number. Let's look at some examples of how they work in the Cartesian space.

In one dimension (a line), vectors are just regular numbers, so adding the vector $2$ to the vector $-3$ is just

\scriptsize2+(-3)=-1