Notes - Linear Algebra I MT22, Chapter 5

Created: December 31, 2022

  • [[Course - Linear Algebra I MT22]]U
  • Content covered:
    • Prove the Steinitz Exchange Lemma (The. 127).
    • Prove that if $V$ is a vector space and $S,T$ are finite subsets of $V$, then if $S$ is linearly independent and $T$ spans $V$, then $ \vert S \vert \le \vert T \vert $. (The. 128)
    • Prove all bases are finite and of the same size. (Cor. 129)
    • Prove if a vector space has a finite spanning set, then it contains a basis (Prop. 132).
    • Prove if $U \le V$, $\dim U \le \dim V$ and if $\dim U = \dim V$ then $U = V$ (Prop. 134).
    • Prove a superset of a linearly independent set is a basis for a vector space. (Prop. 135).
    • Prove a maximal linearly independent subset of a finite-dimensional vector space is a basis. (Cor. 136).
    • Prove a minimal spanning subset of a finite-dimensional vector space is a basis. (Cor. 137)
    • Prove the dimension formula. (The. 139)


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