Lecture11: 4 subspaces & transformations + transformers

Four Subspaces and Their Transformations

Unraveling the connection between vector subspaces and their transformations in the broader context of linear algebra.

Subtopics:

• A brief refresher: The four fundamental subspaces.
• How transformations affect these subspaces.
• Intuition behind transformations in vector spaces.

Transformer Matrices & Their Relationship with Transformers

Discovering the inherent relationship between matrices and the transformations they represent.

Topics Covered:

• Understanding transformer matrices: Definition and properties.
• How these matrices facilitate transformations.
• Real-world implications of such transformations.

Orthogonality of the Four Fundamental Subspaces

A deep dive into the orthogonal relationships among the core subspaces of linear algebra.

Key Insights:

• Revisiting the orthogonal relations among the four subspaces:
  • Rowspace & Null Space
  • Columnspace & Left Null Space
• Significance of orthogonality in linear transformations and solutions.

Least Squares, Linear Regression, and Projections

Understanding the mathematical formulations and practical applications of these intertwined concepts.

Subtopics:

• The core idea behind the least squares method.
• Linear regression: Definition, applications, and its relationship with least squares.
• Projections in the context of least squares and linear regression.