Linear Algebra: Quadratic Forms and Matrix Diagonalization

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Revisits the spectral theorem for symmetric matrices, emphasizing orthogonally diagonalizable properties and its equivalence with symmetric bilinear forms.

Differential Forms on Manifolds

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.

Vector Spaces: Properties and Examples

Explores vector spaces, focusing on properties, examples, and subspaces within a practical exercise on polynomials.

Vector Spaces: Definitions and Applications

Introduces vector spaces, subspaces, linear maps, and evaluation maps, with examples and exercises for better comprehension.

Quadratic Forms: Definitions, Examples

Covers the definition of quadratic forms in R^n with examples in R^2 and R^3.

Vector Subspaces in R4

Explores vector subspaces in R4, symmetric matrices, basis vectors, and canonical forms.

Linear Algebra Basics: Vector Spaces, Transformations, Eigenvalues

Covers fundamental linear algebra concepts like vector spaces and eigenvalues.

Linear Transformations: Polynomials and Bases

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

Symmetric Matrices: Diagonalization

Explores symmetric matrices, their diagonalization, and properties like eigenvalues and eigenvectors.

Diagonalization of Linear Transformations

Explains the diagonalization of linear transformations using eigenvectors and eigenvalues to form a diagonal matrix.