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Vector Spaces

Theorem

All Bases have same size

Theorem

Cauchy-Schwarz Inequality

Theorem

Complement of Subspace Generates Direct Sum

Theorem

Criterion for a Basis

Theorem

Criterion for Direct Sum

Theorem

Criterion for Subspace

Theorem

Dependence Lemma

Theorem

Double Complement Returns the Original Subspace

Theorem

Every Spanning Set Contains a Basis

Theorem

Isomorphic Vector Spaces have the Same Dimension

Theorem

Linear Independent Sets are Smaller than Spanning Sets

Theorem

Linearly Independent Sets Generate Bases

Theorem

Orthogonal Decomposition

Theorem

Parallelogram Equality

Theorem

Properties of Orthogonal Complements

Theorem

Span of Vectors is a Subspace of the Vector Space

Theorem

Sum of Subspaces is a Subspace

Theorem

Sum of Subspaces is Smallest Subspace Containing their Union

Theorem

The Orthogonal Complement is a Complementary Subspace