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Ch01 Ch03

Posted on September 19, 2026 by pingzhou.yuan1@gmail.com
Table of Contents (click to expand)
  • Ch01 – Vector Spaces
    • Subspace
      • Sum of Subspaces
      • Direct Sums
  • Ch02 – Finite-Dimensional
    • Span and Linear Independence
      • Linear Combinations and Span
      • Linear independence
    • Bases
    • Dimension
  • Ch03 – Linear Maps
    • Vector Space of Linear Maps
    • Null Space and Ranges
    • Matrices
      • Matrix of a linear map
      • Column-row Factoring
    • Invertibility and Isomorphisms
      • Invertible Linear Maps
      • Isomorphic Vector Spaces
      • Linear Map VS Matrix-vector Multiplication
      • Change of Basis
    • Products and Quotients of Vector Spaces
      • Products of Vector Spaces
      • Quotient Spaces
    • Duality
      • Dual Space and Dual Map
      • Null Space and Range of Dual of Linear Map
      • Matrix of Dual of Linear Map

Subspace

Definition 1.0 – Subspace of Vector Space

A subspace $U$ of a vector space $V$ is a subset of $V$ that is itself a vector space under the same addition and scalar multiplication as $V$. Formally, it has two properties:

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