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1. Systems of linear equations.

1.1. Notion of systems of linear equations.

1.2. Gaussian elimination.

1.2.1. Matrix notation.

1.2.2. Row reduction and echelon form.

1.3. Homogenous linear systems.

1.4. Applications.

2. Matrices and determinants.

2.1. Matrices.

2.1.1. Operation with matrices.

2.1.2. The inverse of a matrix.

2.1.3. Partitioned matrices.

2.1.4. LU factorization.

2.2. Determinants.

2.2.1. Properties.

2.2.2. Cramer's rule.

3. Real vector spaces.

3.1. Vector spaces and subspaces.

3.2. Null spaces and column spaces.

3.2.1. Linear transformation.

3.3. Linearly independent sets. Bases.

3.4. Dimension and rank.

3.5. Change of basis.

4. Eigenvalues and eigenvectors. Diagonalization.

4.1. Eigenvalues and eigenvectors.

4.2. Diagonalization.

5. Orthogonality and least-square problems.

5.1. Inner product, length, orthogonality.

5.2. Orthogonal projections.

5.3. The Gram-Schmidt process.

5.4. The least-squares problem.

6. The singular value decomposition.

6.1. Symmetric matrices.

6.2. Singular value decomposition.

6.3. Moore-Penrose pseudoinverse matrix.

6.4. Applications to least-square problem.