Gram-Schmidt example with 3 basis vectors | Linear Algebra | Khan Academy

TL;DR

Gram-Schmidt process is used to find an orthonormal basis for a given subspace.

Transcript

Let's do one more Gram-Schmidt example. So let's say I have the subspace V that is spanned by the vectors-- let's say we're dealing in R4, so the first vector is 0, 0, 1, 1. The second vector is 0, 1, 1, 0. And then a third vector-- so it's a three-dimensional subspace of R4-- it's 1, 1, 0, 0, just like that, three-dimensional subspace of R4. And w... Read More

Key Insights

• 😫 The Gram-Schmidt process is a method used to transform any set of linearly independent vectors into an orthonormal basis.
• ❓ By replacing vectors with orthogonal versions and normalizing them, the Gram-Schmidt process simplifies calculations involving linear combinations and projections.

Questions & Answers

Q: What is the purpose of the Gram-Schmidt process?

The Gram-Schmidt process is used to find an orthonormal basis for a given subspace, making it easier to work with linear combinations and computations.

Q: How is the length of a vector calculated?

The length of a vector is calculated using the Euclidean norm, which is the square root of the sum of the squares of its components.

Q: Why is it necessary to normalize the vectors in the Gram-Schmidt process?

Normalizing the vectors ensures that they have a length of 1, making them orthonormal and simplifying further calculations involving linear combinations.

Q: How are the orthogonal vectors obtained in the Gram-Schmidt process?

Orthogonal vectors are obtained by subtracting the projection of a given vector onto the subspace spanned by the previously orthogonalized vectors.

Summary & Key Takeaways

• The video demonstrates the Gram-Schmidt process to find an orthonormal basis for a given subspace in R4.

• The given subspace is spanned by the vectors (0, 0, 1, 1), (0, 1, 1, 0), and (1, 1, 0, 0).

• The process involves replacing each vector with an orthogonal version and normalizing the resulting vectors.