How to Apply the Gram-Schmidt Orthonormalization Process Step-by-Step

📚 What is the Gram-Schmidt Orthonormalization Process?

The Gram-Schmidt process is a method for orthonormalizing a set of vectors in an inner product space. That is, it transforms a set of linearly independent vectors into a set of orthonormal vectors which span the same subspace. This process is fundamental in linear algebra and has applications in various fields like numerical analysis, quantum mechanics, and signal processing.

📜 A Brief History

The process is named after Jørgen Pedersen Gram and Erhard Schmidt. Gram published the method in 1883, while Schmidt presented it independently in 1907. The core idea is to sequentially project each vector onto the orthogonal complement of the subspace spanned by the previously orthonormalized vectors.

🔑 Key Principles

At its heart, the Gram-Schmidt process relies on the following principles:

• 📐 Linear Independence: The initial set of vectors must be linearly independent. This means no vector in the set can be expressed as a linear combination of the others.
• ➕ Vector Projection: Projecting a vector onto another allows us to decompose it into components parallel and perpendicular to that other vector. This is key for creating orthogonal vectors.
• ➗ Normalization: Normalizing a vector involves scaling it to unit length. This creates orthonormal vectors, which are both orthogonal and have a magnitude of 1.

🪜 Step-by-Step Guide to the Gram-Schmidt Process

Let's say we have a set of linearly independent vectors {$v_1, v_2, ..., v_n$}. Here’s how to apply the Gram-Schmidt process:

1. ✨ Step 1: Orthonormalize the first vector

   Let $u_1 = v_1$. Normalize $u_1$ to get $e_1 = \frac{u_1}{||u_1||}$. Here, $||u_1||$ represents the magnitude (or length) of $u_1$.

2. ➕ Step 2: Orthogonalize the second vector

   Let $u_2 = v_2 - \text{proj}_{e_1}v_2 = v_2 - (v_2 \cdot e_1)e_1$. This subtracts the projection of $v_2$ onto $e_1$ from $v_2$, resulting in a vector $u_2$ that is orthogonal to $e_1$.

3. 📏 Step 3: Normalize the second vector

   Normalize $u_2$ to get $e_2 = \frac{u_2}{||u_2||}$. Now, $e_1$ and $e_2$ are orthonormal.

4. 🔄 Step 4: Repeat for the remaining vectors

   For any vector $v_i$ (where $i > 2$), orthogonalize it by subtracting its projections onto all previously found orthonormal vectors:

   $u_i = v_i - \text{proj}_{e_1}v_i - \text{proj}_{e_2}v_i - ... - \text{proj}_{e_{i-1}}v_i = v_i - (v_i \cdot e_1)e_1 - (v_i \cdot e_2)e_2 - ... - (v_i \cdot e_{i-1})e_{i-1}$

5. ✨ Step 5: Normalize the result

   Normalize $u_i$ to get $e_i = \frac{u_i}{||u_i||}$.

6. ✅ Step 6: Continue until all vectors are orthonormalized.

   Repeat steps 4 and 5 until all vectors in the original set {$v_1, v_2, ..., v_n$} have been processed.

💡 Real-world Example

Consider the vectors $v_1 = (1, 1, 0)$ and $v_2 = (1, 0, 1)$. Let's apply the Gram-Schmidt process.

1. 🌱 Step 1: Orthonormalize $v_1$

   $||v_1|| = \sqrt{1^2 + 1^2 + 0^2} = \sqrt{2}$. So, $e_1 = \frac{(1, 1, 0)}{\sqrt{2}} = (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0)$.

2. ➖ Step 2: Orthogonalize $v_2$

   $v_2 \cdot e_1 = (1, 0, 1) \cdot (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0) = \frac{1}{\sqrt{2}}$. Then, $\text{proj}_{e_1}v_2 = (v_2 \cdot e_1)e_1 = \frac{1}{\sqrt{2}}(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0) = (\frac{1}{2}, \frac{1}{2}, 0)$.

   Thus, $u_2 = v_2 - \text{proj}_{e_1}v_2 = (1, 0, 1) - (\frac{1}{2}, \frac{1}{2}, 0) = (\frac{1}{2}, -\frac{1}{2}, 1)$.

3. ✨ Step 3: Normalize $u_2$

   $||u_2|| = \sqrt{(\frac{1}{2})^2 + (-\frac{1}{2})^2 + 1^2} = \sqrt{\frac{1}{4} + \frac{1}{4} + 1} = \sqrt{\frac{3}{2}} = \frac{\sqrt{6}}{2}$.

   So, $e_2 = \frac{u_2}{||u_2||} = \frac{(\frac{1}{2}, -\frac{1}{2}, 1)}{\frac{\sqrt{6}}{2}} = (\frac{1}{\sqrt{6}}, -\frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}})$.

Therefore, the orthonormal basis derived from $v_1$ and $v_2$ is {$(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0), (\frac{1}{\sqrt{6}}, -\frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}})$}.

🤔 Conclusion

The Gram-Schmidt process provides a systematic way to create orthonormal bases from any set of linearly independent vectors. Mastering this process is crucial for understanding many concepts in linear algebra and its applications. With a solid understanding and practice, you'll be able to apply this powerful tool effectively. Keep practicing, and you'll become a pro in no time!