Linear Algebra

1. Vector Space

• Vector Space
• Geometric vectors
• Components of a Vector
• Subspace
• Sums of Subspaces; Direct sum
• Spanning Sets and Linear Independence; Exchange Lemma
• Bases
• Spazi vettoriali finitamente generati
• Euclidean Spaces, ℝⁿ
• The Dimension of a Vector Space
• Formula di Grassmann
• Direct Sum of Subspaces

2. Scalar product and Orthogonality

• Dot product
• Inner Product
• Norms and Isometries
• Hermitian dot product
• Orthogonal space
• Gram Schmidt Orthogonalization Process
• Cross product
• Scalar triple product
• The Levi-Civita symbol

3. Matrices

• Matrices
• Matrix multiplication
• Transpose of a Matrix
• Inverse Matrix
• Rank of a matrix
• Eigenvalues and Eigenvectors
• The Kronecker Product

4. Determinants

• Determinante of a square matrix
• Laplace Theorem
• Basic properties of the determinant; Binet's theorem
• Orthogonal matrix

5. System of linear equations

• Systems of Linear equations
• Homogeneous systems
• Rouché-Capelli Theorem
• Gauss Elimination Method

6. Linear Transformation

• Linear Transformations
• Matrix Representation Theorem
• Image and Kernel of a linear transformation
• Change of basis
• Invertible linear transformations
• Addition and scalar multiplication of linear maps: Hom(V, W).
• Isomorphisms
• Trasformazioni iniettive e suriettive

6. Diagonalization Autovalori e autovettori.

• Diagonalization
• Eigenvalues and Eigenvectors
• Diagonalization criteria
• Change of Basis

8. Linear Forms

• Linear forms
• The Dual Space
• Bilinear Forms
• Multilinear Transformations

9. Tensors

• Tensors
• Tensor definition
• Quadratic Forms