What is the usage of the Gram Schmidt process?

The Gram-Schmidt process is a method to convert linearly independentVectors that cannot be written as a linear combination of each other. vectors to an orthonormal set of vectors. It is widely used in various areas of mathematics to understand the projection of vectors and simplify their computations.

To understand this process, we must first understand the concept of orthonormal vectors.

What are orthonormal vectors?

Orthonormal vectors are defined as a combination of orthogonal and normalized vectors. Let's understand these individually.

• Orthogonal vectors are perpendicular to each other, and their dot product equals zero. Suppose we have two orthogonal vectors: $v_1$ and $v_2$. Their dot product can be represented as:

Note: The angle between orthogonal vectors is $90^o$.

• A normalized vector is a unit vector form of a vector. Suppose we have a vector $v$. Its normal form can be represented as:

Here$||v||$represents the magnitude of the vector.

Note: The magnitude of a normalized vector is $1$.

Gram-Schmidt process

Before diving into the process, let's revise the concept of projection of a vector on another vector to understand the process better.

Projection of a vector on another vector

Projection of a vector onto another vector shows how much of the first vector points in the same direction as the second vector.

Suppose we have two vectors: $v_1$ and $v_2$. We represent projection of vector $v_1$ on $v_2$ as:

Mathematically, we can represent this as:

• $<v_1,v_2>$ : Dot product between vectors $v_1$ and $v_2$.

• $||v_2||^2$ : Square of the magnitude of $v_2$.

Steps of Gram-Schmidt process

There are two steps involved in the process:

1. First, we find the orthogonal vectors for our given set of vectors.

2. We find the unit vectors( normalized form) of the orthogonal vectors.

1) Finding orthogonal vectors

For the given set of vectors $v_i$, we represent the orthogonal vectors as $u_i$where $i=1,2,3..$so on.

We represent this mathematically as:

• For the first orthogonal vector where $i=1$ the expression becomes:

• Similarly, we find the third orthogonal vector where $i=3$ using the expression:

In such a way, we find all the orthogonal vectors for our given vectors by simplifying the expression accordingly.

2) Finding normalized vectors

For the orthogonal vectors $u_i$calculated above, we represent the normal vectors as $e_i$where $i=1,2,3..$so on. We represent this mathematically as:

• For the first normalized vector where $i=1$ the expression becomes:

Example

Let's solve an example using the Gram-Schmidt process now to calculate the orthonormal vectors for given vectors. Suppose we have two vectors such that:

$v_1 = (1,1,1)$

$v_2 = (0,1,1)$

1. Calculating orthogonal vectors:

• Finding $u_1$:

• Finding $u_2$:

• We first calculate $proj_{u_1}(v_2)$. This is the projection of $v_2$ onto $u_1$.

Here, we first take the ratio of the dot product of the two vectors ($v_2$ and $u_1$) and the magnitude of the vector $u_1$. Then we multiply the result with vector $u_1$:

• Now finding $u2$:

We have calculated $u_1$ and $u_2$ and both these vectors are orthogonal to each other.

2. Calculating normalized vectors:

• Finding $e_1$by taking the ratio of $u_1$ with its magnitude:

• Finding $e_2$by taking the ratio of $u_2$ with its magnitude:

We converted our given vectors ($v_1$ and $v_2$) to orthonormal vectors($e_1$ and $e_2$).

Conclusion

The Gram-Schmidt process is a valuable tool in linear algebra to transform vectors into orthogonal or orthonormal sets. This results in the simplification of vector calculation and is crucial in solving systems of equations, constructing bases, and improving signal processing.