Definitions
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Subspace

Sum of subspaces

Direct sum of subspaces

Linear combinaton

Span of a list of vectors

Finite and infinite-dimensional spaces

Linearly dependent and linearly independent lists of vectors

Basis

The dimension of a finite dimensional space

Linear map

Null space of an operator

Range of an operator

The matrix of a linear map

The matrix of a vector

Operator

Injectivity, surjectivity, invertibility of an operator

Inverse map

Isomorphic spaces

Invariant subspaces

Eigenvalues and eigenvectors


Polynomials of operators (operator polynomials)

Matrix of an operator

Upper-triangular and diagonal matrices

Eigenspaces

Diagonalizable operators

Inner product with the properites required by the definition

Norm (induced by the inner product)

Orthogonal decomposition

Orthonormal bases

Projection, orthogonal projection

Gram-Schmidt procedure

Orthogonal compliment

Linear functional

Adjoint (linear map)

Self-adjoint operator

Normal operator

Positive operator

Isometry

Polar decomposition

Singular value decomposition

Generalized eigenvector

Generalized eigenspace

Geometric and algebraic multiplicities of an eigenvalue


Things that require proofs (theorems, propositions, lemmas)
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Theorems:

3.5, 3.22, 3.65, 3.69

5.10, 5.26,  5.41

6.15, 6.18, 6.31, 6.37, 6.51

7.7, 7.15, 7.20,  7.24, 7.27, 7.36, 7.51

8.5