Stanford ENGR108: Introduction to Applied Linear Algebra | 2020 | Lecture 16-VMLS Gram Schmidt algo. | Summary and Q&A

TL;DR

The Gram Schmidt Algorithm is an algorithm used to determine linear independence or linear dependence of a set of vectors.

Key Insights

• 📛 The Gram Schmidt Algorithm is named after mathematicians Graham and Schmidt.
• 😫 Its main purpose is to determine linear independence or linear dependence of a set of vectors.
• 😫 The algorithm generates a set of orthonormal vectors by processing the vectors one at a time.
• ❓ While not necessary to implement it oneself, understanding the algorithm is important.
• ❓ Early termination of the algorithm indicates linear dependence.
• 💾 The complexity of the algorithm is approximately 2nk^2 flops.
• 🈸 The algorithm is useful in various practical applications beyond linear independence.

Transcript

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Questions & Answers

Q: What is the main purpose of the Gram Schmidt Algorithm?

The main purpose of the Gram Schmidt Algorithm is to detect whether a set of vectors is linearly dependent or linearly independent.

Q: Are students expected to implement the algorithm themselves?

While it is not necessary for students to implement the algorithm themselves, it is important for them to understand how it works and what it does. In practice, one would rely on pre-existing implementations.

Q: What happens if the algorithm terminates early?

If the algorithm terminates early, it means that the set of vectors being processed is linearly dependent.

Q: What is the complexity of the Gram Schmidt Algorithm?

The complexity of the Gram Schmidt Algorithm is approximately 2nk^2 flops, where n is the dimension of the vectors and k is the number of vectors being processed.

Summary & Key Takeaways

• The Gram Schmidt Algorithm is used to detect whether a set of vectors is linearly dependent or linearly independent.

• It is an algorithm used to process vectors one by one and generate a set of orthonormal vectors.

• The algorithm can be used in various practical applications but is mainly focused on determining linear independence.