The main aim of this course is to familiarize the student with the vector spaces, the linear transformations, the matrix representation of a linear transformation and the diagonalization of a linear transformation.
Pedagogical Objectives: At the end of this course, the student should be able to :
1. Recognize a vector space and a subspace of a given space .
2. Recognize that a vector space is a direct sum of 2 or several subspaces.
3. Recognize a free family, a generating family, a basis in a vector space.
4. Recognize the dimension of a vector space.
5. Know and use the formulas with dimensions concerning the subspaces.
6. Know and use the formulas with dimensions concerning dimensions of the image and the kernel.
7. Know the quotient space .
8. Know the structure of the vector space LF (V,W).
9. Represent an element in a vector space or a linear transformation in a matrix form relative to an ordered basis.
10. Determine the transition matrix from an ordered basis to another.
11. Know and use the formula of change of bases.
12. Know and determine the diagonalization of a linear transformation
Vector spaces: definition and examples, subspaces, sum and direct sum, linearly independent, spanning set, basis, dimension.
Linear transformation: definition, range, kernel, dimension formula, operation on the linear transformation.
Matrix and linear transformation: matrix representation, isomorphism, transition matrices, rank and minor.
Operator reduction: eigenvalues and eigenvectors, characteristic polynomial, diagonalization.
Comment: It is advised to start from an example which the students know well that is the real vector space V of the vectors of the space.One will insist on the fact that a vector space is a generalization. Each time the teacher proposes to introduce a new concept (subspace, free subset, spanning subset, basis, linear transformation…) it can start by showing its origin in V.
One will multiply the examples of a vector space in particular in Algebra (set of matrices, set of real polynomials,…) and in Analyzis ( set of functions defined on an interval, set of real series,…)
A study in parallel of the free and generating families is very useful to arrive so that a maximal free family is a basis and a minimal generating family is a basis.
One will insist on the fact that to define a linear mapping, it is enough to determine the images of the elements of a basis of the starting space .
The vector spaces in this course are all finite dimensional.
It is necessary to study the matrix representation of a linear mapping , the transition matrix and the diagonalization of a linear transformation.
It is advised to give examples that the students know well.