Quadratic Forms: Definitions, Examples

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Explores quadratic forms, symmetric bilinear forms, and their properties.

Covers linear transformations between polynomial spaces and explores examples of linear independence and bases.

Explores polynomials with real coefficients, complex roots, and sequences' properties.

Explores quadratic forms in IR³, matrix properties, diagonalization, and positive definite matrices.

Explores Sylvester's Inertia Theorem, relating eigenvalues to diagonal entries in symmetric matrices.

Introduces linear combinations in vector spaces, operations, and polynomials of degree 2.

Explores pseudo-Euclidean spaces, emphasizing isometries and bases in vector spaces with non-degenerate quadratic forms.

Explores polynomial operations, properties, and subspaces in vector spaces.

Covers non-negative definite matrices, covariance matrices, and Principal Component Analysis for optimal dimension reduction.

Introduces abstract concepts in linear algebra, focusing on operations with vectors and matrices.