1. Vector Space

Vector Space \( \forall u,v \in V \)

2. Subspace

A subspace is a subset of a vector space that is itself a vector space under the same operations.

3. Dimension

The dimension of a vector space is the number of vectors in its basis.

4. Matrix

A matrix is a rectangular array of numbers that can represent linear transformations.

5. Product and Quotient

The product and quotient of vector spaces are important constructions in linear algebra.

6. Dual Space

The dual space of a vector space consists of all linear functionals on that space.

7. Polynomial

Polynomials can be studied as elements of a vector space.

8. Invariant Subspace

An invariant subspace is a subspace that is mapped to itself by a linear operator.

9. Eigenvalue and Eigenvector

Eigenvalues and eigenvectors are fundamental to understanding linear transformations.

10. Normed Vector Space

A normed vector space is a vector space equipped with a norm.

11. Orthonormal Basis

An orthonormal basis is a basis where all vectors are orthogonal and of unit length.

12. Self-adjoint and Normal

Self-adjoint and normal operators have special properties in linear algebra.

13. Spectral Theorem

The spectral theorem describes the diagonalization of certain types of operators.

14. Isometries

Isometries are linear transformations that preserve distances.

15. SVD

Singular Value Decomposition (SVD) is a factorization of a matrix.

16. Generalized Eigenvector & Nilpotent

Generalized eigenvectors and nilpotent operators are used in the study of non-diagonalizable matrices.

17. Characteristic Polynomial

The characteristic polynomial of a matrix encodes information about its eigenvalues.

18. Jordan Normal Form

The Jordan normal form is a canonical form for matrices over an algebraically closed field.

19. Complexification

Complexification extends real vector spaces to complex vector spaces.

20. Trace & Determinant

The trace and determinant are important invariants of a matrix.