021a MOC Linear Algebra

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Topics

Vector spaces

• Vector Space
  • Vector Space Norm
    • Normed Vector Space
    • Every inner product induces a norm in a vector space
    • Every norm induces a metric function in a vector space

Span Properties

• Spanning Set
  • Fields can be seen as one-dimensional vector spaces over themselves
  • Finitely Generated Vector Space
  • Steinitz Exchange Lemma: Every linearly independent set of a finitely generated vector space has at most the same number of vectors as its spanning set
    • All Hamel bases have the same number of elements (in a finitely generated vector space)

Linear (In)Dependence Properties

• Linear Independence
  • Linear Dependence
  • The union of a linearly independent set with a vector outside of its span is also linearly independent
    • Any linearly independent set can be extended to a Hamel basis
  • A set is linearly dependent iff there is a vector which is a linear combination of previous vectors
  • Sets larger than a finitely generated space’s bases are linearly dependent
  • A set is linearly independent iff all its finite subsets are also L.I.

Hamel Basis Properties

• Hamel Basis
  • Every finitely generated vector space has a Hamel basis
  • All Hamel bases have the same number of elements (in a finitely generated vector space)
  • Every vector space which has a linearly independent set has a Hamel Basis containing this set
• Finitely Generated Vector Space Dimension
  • Every vector space with same dimension are isomorphic to each other

Linear Transformation Properties

• Linear Transformation
  • A linear transformation is injective iff its kernel is trivial
  • Kernel-Image Theorem
  • Vector Space Isomorphism
  • Endomorphism
  • An endomorphism is injective iff it’s surjective
• Diagonalization
• Trace
• Determinant
• Positive Definiteness

Dual Space Properties

• Vector Space Dual
• Linear Operator

Related

• 022 MOC Functional Analysis
• Affine Space
  • Affine Map
  • Euclidean Affine Space