Orthonormal Bases: Definition & Example

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson we show how independent vectors in a space can become a basis for the space and how this basis can be turned into an orthonormal basis. Having an orthonormal basis is useful in many applications involving vectors.

The Game Plan

Having an overview, a game plan, for an event helps to keep us on track. The same is true when describing the math in this lesson.

We will start with the idea of a basis as being a collection of linearly independent vectors which span a space. The space we will use is the four-dimensional space called R4. We will find another basis for R4 where the vectors are orthogonal to each other. Then, we will normalize each of these orthogonal basis vectors to produce a basis in R4 where each of the four basis vectors is orthogonal to each other and each basis vector has a unit length. This basis is called an orthonormal basis. To represent any arbitrary vector in the space, the arbitrary vector is written as a linear combination of the basis vectors. Having an orthonormal basis greatly simplifies calculating the coefficients for the linear combination. Finally, we state the form of the standard orthonormal basis.

Spanning a Space

We start with the following four vectors: b1, b2, b3 and b4:


b1=[1,1,0,0],b2=[1,1,0,0],b3=[0,1,1,0],b4=[0,0,0,1]


Note: to make calculations simpler, the numbers chosen in this example are 1s and 0s. Any numbers are potentially good numbers to use.

You might be wondering if these ''vectors'' are the same vectors routinely used in discussions of force, acceleration and velocity. They are! But the physics and engineering vectors we usually use are for the three-dimensional space we live in where the basis vectors are i = [ 1, 0, 0], j = [ 0, 1, 0] and k = [ 0, 0, 1]. In this lesson, we are generalizing to higher dimensions. Same ideas, just a little more abstract. Any space with a finite dimension can have basis vectors. The requirements for a basis are:

  • the basis vectors are linearly independent
  • the basis vectors span the space

What about our four b vectors? Do they span R4? This asks if we can write any vector in the space as a linear combination of the basis vectors:


x=c1b1+c2b2+c3b3+c4b4


For example, if our arbitrary vector is x = [ 2, -3, 15, -6], then


x=-7[1,1,0,0]+4[1,1,0,0]+11[0,1,1,0]-6[1,0,0,1]


Let's check:


-7[1,1,0,0]+4[1,1,0,0]+11[0,1,1,0]-6[1,0,0,1]=[2,-3,15,-6]


Multiplying the components of each vector by the coefficient in front of the vector:


[-7,-7,0,0]+[4,4,0,0]+[0,11,11,0]+[-6,0,0,-6]=[2,-3,15,-6]


Adding the components:


-7[1,1,0,0]+4[1,1,0,0]+11[0,1,1,0]-6[0,0,0,1]=[2,-3,15,-6]


And the result is x!

There are an infinite number of possible choices for a set of basis vectors. The only requirements are linear independence and the ability to span the space.

Orthogonalizing a Basis

Visualizing in four dimensions is difficult. What if we looked at only three of the independent vectors in three-dimensional space?


These independent basis vectors span the space.
These_independent_basis_vectors_span_the_space


Now, after orthoganalizing the basis:


The basis vectors are now perpendicular to each other.
The_basis_vectors_are_now_perpendicular_to each_other


Let's get back to our example in 4 dimensions. The b vectors are the independent vectors forming a basis. Orthogonal vectors are at a right angle to each other. They are perpendicular. To turn our basis of independent vectors into a basis of orthogonal vectors, we select one of the vectors. For example, choose b1. We rename this vector as g1. Then we find the part of the second vector, b2, which is perpendicular to g1. The result is called g2. The process, called the Gram-Schmidt process, continues until we have a new set of vectors where each vector is perpendicular to each other vector. The resulting orthogonal basis:


g1=[1,1,0,0],g2=[0,0,1,0],g3=[1/2,-1/2,0,0],g4=[0,0,0,1]


How do we know these vectors are perpendicular to each other? We use the dot product (also called the inner product). Remember from physics, the dot product of two vectors is the product of the length of each vector multiplied by the cosine of the angle between the vectors. For perpendicular vectors, the angle is 90o and the cosine of 90o is 0. Great! But how do we take the dot product of two vectors like g1 with g2?


g1.g2=[1,1,0,0].[0,0,1,0]=1(0)+1(0)+0(1)+0(0)=0


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