What is the Gram-Schmidt process?

We use the Gram-Schmidt process to calculate an orthogonal basis $\vec{u_{1}}, \vec{u_{2}}, ..... , \vec{u_{n}}$ that spans the same space from the given set of linearly independent vectors $\vec{v_{1}}, \vec{v_{2}}, ..... , \vec{v_{n}}$.

Background

Before getting started with the Gram-Schmidt process, here are a few things that we need to recall:

Note: An orthogonal set contains non-zero vectors $\vec{v_{1}}, \vec{v_{2}}, ..... , \vec{v_{n}} \in V$ . Both are orthogonal to each other—$\langle \vec{v_{i}} , \vec{v_{j}} \rangle = 0$ for $i \neq j$.

Process

Suppose the arbitrary set of linearly independent vectors $\vec{v_{1}}, \vec{v_{2}}, ..... , \vec{v_{n}} \in V$ span $R^n$. To calculate its orthogonal basis $\vec{u_{1}}, \vec{u_{2}}, ..... , \vec{u_{n}}$ that spans the same space, we perform the following steps:

Step 1

We calculate the first vector, $\vec{u_{1}}$in the following way:

Step 2

We calculate the second vector, $\vec{u_{2}}$, in the following way:

Step 3

We calculate the third vector, $\vec{u_{3}}$, in the following way:

Step 4

We calculate the $n^{th}$ vector, $\vec{u_{n}}$, in the following way:

The vectors $\vec{u_{1}}, \vec{u_{2}}, ..... , \vec{u_{n}}$ form the orthogonal basis for $R^n$.

Example

Find an orthogonal basis for the following vectors that span $R^3$:

Step 1

The vector $\vec{u_{1}}$ equals the following:

Step 2

The vector $\vec{u_{2}}$ equals the following:

We get the following equation upon calculation:

Step 3

The vector $\vec{u_{3}}$ equals the following:

We get the following equation upon calculation:

Solution

The following vectors have been calculated using the Gram-Schmidt process that forms the orthogonal basis for $R^3$:

Orthonormal basis

Using the Gram-Schmidt process, an orthonormal basis $x_{1}, x_{2}, ...., x_{n}$ can be calculated from any given set of linearly independent vectors.

Note: An orthonomal set has non-zero vectors $\vec{x_{1}}, \vec{x_{2}}, ..... , \vec{x_{n}} \in X$ with a norm equal to one—$||x_{i}|| = 1$.

Procedure

To calculate an orthonormal basis, we follow the steps below:

• We convert the given set of vectors into an orthogonal basis using the Gram-Schmidt process.

• We divide the resulting vectors by their norm to achieve an orthonormal basis.

Example

Let's use the previous example:

The orthogonal set of vectors achieved for $V$ is as follows:

The orthonormal vectors will be calculated in the following way:

Quiz

To help you completely grasp the Gram-Schmidt process, here are a few questions with solutions: