Lecture09: 4 fundamental subspaces & proof for dim(rowspace)=dim(colspace)

Introduction to the Four Fundamental Subspaces

Every matrix presents four distinct subspaces that offer various perspectives on its structure and information.

The Big Four:

• Rowspace
• Columnspace
• Null Space (or Kernel)
• Left Null Space (Null space of ( A^T ))

The Null Space: Beyond the Basics

Taking our understanding of the null space a step further, exploring its characteristics and significance.

Characteristics:

• Intuitive understanding: What vectors are in the null space?
• Computational methods: How to deduce the null space for complex matrices?

Ranks & The Four Subspaces

Delving deep into the concept of rank, especially in the context of these core subspaces.

Rank Definitions:

• Rank of a matrix: Maximum number of linearly independent column vectors in a matrix (or rows).
• Connection between rank and the null space
• The role of rank in determining matrix properties

Basis & Independence: The Cornerstones

Understanding the foundational concepts that underpin our entire exploration of subspaces.

Topics Explored:

• What does it mean for vectors to be linearly independent?
• How is the basis of a subspace determined?
• Basis of each of the four fundamental subspaces

Proving the Dimensionality Equality

A rigorous proof to show that the dimension of the rowspace is equal to the dimension of the columnspace for any given matrix.

Proof Overview:

• Starting with the basics: Definitions and axioms
• Leveraging matrix properties and rank
• Concluding with the equality: dim(rowspace) == dim(colspace)