Lecture13: QR Decomposition, Least Squares, Gram-Schmidt

4 Subspaces & Their Transformations

Revisiting the fundamental subspaces of linear algebra and understanding how they undergo transformations.

Subtopics:

• Basics of the four primary subspaces.
• How transformations impact each subspace.

Least Squares & Linear Regression

Consolidating our grasp on how projections influence linear regression and the pivotal role of the least squares method.

Key Concepts:

• Least Squares method: How it optimizes linear regression models.
• Practical applications and real-world implications.

QR Decomposition: An Introduction

Venturing into the world of QR decomposition, an essential technique in numerical linear algebra.

Highlights:

• The essence of QR decomposition and its significance.
• Steps in QR decomposition and its properties.

Orthogonal Matrices: A Deep Dive

Understanding the unique properties of orthogonal matrices and their role in various mathematical processes.

Topics Covered:

• Definition and properties of orthogonal matrices.
• Significance and use-cases of orthogonal matrices in linear algebra.

Gram-Schmidt Process

Ending with a practical and essential process to generate an orthonormal basis for any subspace.

Subtopics:

• The algorithm behind the Gram-Schmidt process.
• Applications and significance in real-world problems.