# Linear Algebra ME

### Course topics

- The real numbers, inequalities in real numbers, the complex numbers, the Cartesian representation, the polar representation, the exponential representation, the Theorem of de Moivre, root computations.
- Systems of linear equations over the real or complex numbers, the solution set and its parametric representation, echelon form and the reduced echelon form of a matrix, backwards substitution, forward substitution and their complexity, the Gauss elimination algorithm and its complexity, the reduction algorithm and its complexity.
- Vector spaces, sub-spaces of vector spaces, linear combinations of vectors, the span of a set of vectors, linear dependence and linear independence, the dimension of a vector space, row spaces and column spaces of matrices, the rank of a matrix.
- Linear mappings between vector spaces, invertible mappings and isomorphisms, the matrix representation of finite dimensional linear mappings, inversion of a square matrix, composition of mappings, multiplication of matrices, the algebra of matrices, the kernel and the image of a linear mapping and the computation of bases, changing of a basis, the dimension theorem for linear mappings.
- Inner product spaces, orthogonality, the norm of a vector, orthonormal sets of vectors, the Cauchy-Schwarz inequality, the orthogonal complement of a sub-space, orthogonal sequences of vectors, the Gram-Schmidt algorithm, orthogonal transformations and orthogonal matrices.
- The determinant of a square matrix, minors and cofactors, Laplace expansions of the determinant, the adjoint matrix and Laplace theorem, conjugation of a square matrix, similarity transformations and their invariants (the determinant and the trace).
- Eigenvalues, eigenvectors, eigenspaces, diagonalization and similarity, the characteristic polynomial, the algebraic and the geometric multiplicities of an eigenvalue, the spectral theorem for Hermitian matrices.

### Course Information

- University course catalogue:
- 201.1.9321
- Level:
- Service
- Credits:
- 4.5

##### Recently Given

- 2020–21–B (Prof. Mikhail Muzychuk)
- 2020–21–A (Prof. Nadya Gurevich)
- 2019–20–B ()
- 2019–20–A ()
- 2018–19–B (Dr. Saak Gabriyelyan)
- 2018–19–A (Prof. Nadya Gurevich)
- 2017–18–B (Dr. Stewart Smith)
- 2017–18–A ()

###### Departments

- Software and information systems engineering
- Geology
- Faculty - Engineering
- Chemical engineering
- Structural engineering
- Industrial engineering
- Mechanical engineering