Linear algebra

Course Title: Linear Algebra

Institution: National Higher School of Artificial Intelligence
Academic Level: First-Year Engneers
Course Duration: One Semester

Course Description:

Linear algebra is the mathematics of dimensions - where lines become planes, numbers become vectors, and equations transform into multidimensional spaces. In this course, you'll master the fundamental language that describes these transformations. We'll explore how systems of equations behave in higher dimensions, discover the hidden structures within matrices, and unlock powerful tools for solving complex problems. Through both theory and practice, you'll develop an intuitive understanding of vector spaces and linear mappings that form the bedrock of advanced mathematics and its applications.

Key Topics Covered:

1. Vectors & Vector Spaces

   • Vector operations, linear independence, basis, and dimension.

   • Subspaces, span, and coordinate systems.

2. Matrices & Matrix Algebra

   • Matrix operations (addition, multiplication, transpose).

   • Inverses, determinants, and special matrices (diagonal, symmetric, orthogonal).

3. Systems of Linear Equations

   • Gaussian elimination, row echelon form.

   • Rank, null space, and solution sets.

4. Linear Transformations

   • Matrix representations, kernel (null space), and image (column space).

   • Change of basis and similarity transformations.

5. Eigenvalues & Eigenvectors

   • Characteristic polynomial, diagonalization..

6. Inner Product Spaces

   • Dot product, orthogonality, projections.

   • Gram-Schmidt process and least squares approximation.

7. Matrix Decompositions (Introductory)

   • LU decomposition, QR factorization.