Università Roma Tre

Bilinear forms: orthogonality, isometries, the adjoint of an operator. Alternate bilinear forms: classification; the symplectic group and the symplectic transvections. Symmetrical bilinear forms: associated quadratic forms, diagonalization theorem; classification over the field of complex numbers or real numbers; the orthogonal group and the symmetries; hyperbolic extensions; Witt's cancellation and extension theorems; anisotropic decomposition. Sesquilinear forms: classification of Hermitian and anti-Hermitian sesquilinear forms. Scalar products: norm and distance (parallelogram law); inequality of Chaucy-Schwarz; the orthogonality operation on subspaces; orthogonal projections and their properties (Bessel's equality, properties of the best approximation); orthogonal and orthonormal sets (Fourier expansion, orthogonal projection formula, Parseval inequality/equality, Gram-Schmitt orthogonalization procedure). Normal operators: self-adjoint operators (Hermitian or symmetric), anti-self-adjoint (anti-Hermitian or anti-symmetric), isometric (unitary or orthogonal), positive and semi-positive. Structure theorem for complex normal operators (or spectral theorem on the field of complex numbers). Structure theorem for real normal operators. The spectral theorem on the field of real numbers. Polar decomposition of an operator. Projective spaces: definition, relationship with affine spaces, duality, projectivities, projective hypersurfaces, the classification of quadratic hypersurfaces.

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S. Roman: Advanced Linear Algebra. Springer, 2008.
S. H. Weintraub: A guide to Advanced Linear Algebra. Mathematical Association of America. 2011.
B. N. Cooperstein: Advanced Linear Algebra. 2nd Edition. Taylor and Francis Group. 2015.
S. Axler: Linear Algebra Done Right. 2nd Edition. Undergraduate Text in Mathematics. Springer. 1997.