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Research

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Theorems

Theorem

All Bases have same size

Theorem

Cauchy-Schwarz Inequality

Theorem

Complement of Subspace Generates Direct Sum

Theorem

Composition Rules for Matrices

Theorem

Conditions for Diagonalizability

Theorem

Criterion for a Basis

Theorem

Criterion for Direct Sum

Theorem

Criterion for Invertibility using Upper Triangular

Theorem

Criterion for Subspace

Theorem

Criterion for Upper Triangular Matrix

Theorem

Dependence Lemma

Theorem

Determinants for Linearly Transformed Autocorrelation Matrices

Theorem

Dimensionality & Linear Maps

Theorem

Distinct eigenvalues have linearly independent eigenvectors

Theorem

Double Complement Returns the Original Subspace

Theorem

Eigendecomposition of a Matrix

Theorem

Eigenvalues are Diagonal Elements of Upper Triangular

Theorem

Enough Diagonal Elements Imply Diagonalizability

Theorem

Every Spanning Set Contains a Basis

Theorem

Existence of Eigenvalues on Complex Spaces

Theorem

Existence of Upper Triangular Matrices on Complex Spaces

Theorem

Finite Variance = Autocorrelation symmetric + positive semidefinite

Theorem

Injectivity, surjectivity, and isomorphism are equivalent when Dimension is the Same

Theorem

Inverse Property of Matrix of Linear Map

Theorem

Inverse to a Linear Map is Unique

Theorem

Isomorphic Vector Spaces have the Same Dimension

Theorem

Isomorphism is a Bijection

Theorem

Jordan Canonical Form

Theorem

Kernel and Image are Subspaces

Theorem

Linear Independent Sets are Smaller than Spanning Sets

Theorem

Linear Map is Injective iff Kernel is 0

Theorem

Linear maps are defined on a basis

Theorem

Linear Maps are Isomorphic to their Matrices

Theorem

Linear Transform for Autocorrelation Matrices

Theorem

Linearity Properties of Matrices

Theorem

Linearly Independent Sets Generate Bases

Theorem

Norm-Preserving Matrix

Theorem

Orthogonal Decomposition

Theorem

Orthogonal Matrices have Determinant 1

Theorem

Parallelogram Equality

Theorem

Positive Definite = Positive Eigenvalues

Theorem

Positive Semidefinite has dim Eigenvectors

Theorem

Properties of Linear Maps

Theorem

Properties of Orthogonal Complements

Theorem

Rank Nullity Theorem

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

Theorem

Trace and Determinant with Eigenvalues