LINEAR ALGEBRA

Vector spaces: Definition, Basis and dimension, Direct sums.

Linear transformations: Definition, Rank-nullity theorem, Algebra of linear transformations, Dual spaces, Matrices.

Systems of linear equations: elementary theory of determinants, Cramer's rule. Eigenvalues and eigenvectors, the characteristic polynomial, the Cayley-Hamilton Theorem, the minimal polynomial, algebraic and geometric multiplicities, Diagonalization, The Jordan canonical form.

Symmetry: Group of motions of the plane, Discrete groups of motion, Finite groups of S0(3).

Bilinear forms: Symmetric, skew symmetric and Hermitian forms, Sylvester’s law of inertia, Spectral theorem for the Hermitian and normal operators on finite dimensional vector spaces.

 

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