#### Objective

#### Content

CHAPTER I: Matrices.
1.1. Introduction: We give the definition of a field and as examples we give the fields , and . Definition of a matrix, entries, main entry, row matrix, column matrix, square matrix, diagonal matrix, diagonal entries, zero matrix and unit matrix, upper triangular matrix, lower triangular matrix and triangular matrix.
1.2. Operations on matrices: addition of matrices and multiplication of a matrix by a scalar and we state their properties without proof. We define the transpose of a matrix and we give its properties. Also we give the definition of an invertible matrix, and we state without proof the theorem which says that if A,BM (K), then AB=I BA=I .
CHAPTER II: Row echelon form of a matrix.
2.1. Row operations: Leading entry of a non-zero row (resp. column), definition of matrix in row echelon form, row operations and elementary row operations, computation of a row echelon form of a matrix, via examples.
2.2. Invertible matrix and row echelon form: We show that the ith row of AB is equal to the product of the ith row of A by B and that the jth column of AB is the product of A by the jth column of B. We prove that if a square matrix is invertible, then every row and every column is non-zero. We admit that if a (nn) matrix A is invertible and B is obtained from A by a finite sequence of row operations, then B is invertible. Also we admit that A is invertible if and only if A can be changed to I by a finite sequence of elementary row operations, and then we give the method that allows to calculate the inverse of a matrix by carrying some appropriate row operations simultaneously on A and I .
CHAPTER III: Determinant.
3.1. Definition and properties: We define the determinant of a matrix by induction on n and we admit the following:
(i) = , A,BM (K).
(ii) = (-1) a +...+(-1) a , 1in.
(iii) = (-1) a +...+(-1) a , 1jn.
Then we state without proofs the properties of determinants and we give examples. We admit that a square matrix is invertible if and only if its determinant is non-zero.
3.2. Rank of a matrix: Definition of a minor, order of a minor, definition of the rank of a matrix as the greatest order of non-zero minors of A. We state without proof that a square matrix of order n is invertible if and only its rank is n, then we admit that rank(A)=rank(t ). Finally we admit without proof that the rank of A is equal to the number of non-zero rows of a row echelon form of A.
CHAPTER IV: System of linear equations.
4.1. Definition and solutions: Definition of linear equation with n variables x ,...,x over K and definition of a system of linear equations. Matrix of a system, matrix representation, definition of solution, principal determinant, principal equations and principal unknowns, characteristic determinants. We admit without proof that a system has solutions if and only if every characteristic determinant of system (I) is zero. As corollary we show that if rank(A)=number of rows of A, then the system has solutions in K.Also if the system has solutions we state without proof the formula giving these solutions, by using the principal determinant and the principal unknowns. We define Cramer’s system and we show that it has a unique solution and we show that a system has a unique solution if and only if it is a Cramer’s system. We prove the important result which states that if AM (K) and if the system A = 0, has a non-zero solution, then = 0.
4.2. Echelon form and system of linear equations: Resolution of a system of linear equations by using the augmented matrix.
CHAPTER V: Vector spaces.
Let K be a field and E be non-empty set.
5.1. Definition and properties. We define an action (or scalar multiplication) of K on E to be every mapping of KE to E. If f is an action of K on E, then the image by f of every element (a,x) of KE is denoted ax. Definition of a vector space over K, examples: M (K), the set K = {(a ,...,a ) ; a ,...,a K}, where (a ,...,a )+(b ,...,b ) = (a +b ,...,a +b ) and (a ,...,a ) = (a ,...,a ). We state the rules of calculations in a vector space.
5.2. Subspace. Definition, examples, intersection of two subspaces, definition of the sum of two subspaces, direct sum.
5.3. System of generators: Let S={x ,...,x } be a subset of E. definition of linear combination of elements of S, the set L(S) of all the linear combinations of elements of S, the properties of L(S), namely:
(i) SL(S), (ii) L(S) is a subspace of E over K.
We give the definition of a system of generators and we state that if x ,...,x E, then x ,...,x form a system of generators of E over K if and only if every element x of W can be written in the form x=a x +...+a x , where a ,...,a K.
CHAPTER VI: Basis of vector space.
6.1. Linear dependence: linearly dependent and linearly independent elements, examples.
6.2. Basis of a vector space. We shall say that {x ,...,x } is a basis of E over K if x ,...,x are linearly independent and form a system of generators of E, examples the canonical bases of M (K), M (K), K and K . dimension of a vector space. We admit that if dim (E)=n, with n0, then
if x ,...,x are elements of E linearly independent over K, then {x ,...,x } is a basis of E.
CHAPTER VII: Reduction of matrices.
Characteristic polynomial of a matrix, similar matrices, eigenvectors and eigenvalues of a matrix, the space
V (A), diagonalization of a matrix.Computation of A when A is diagonalizable.
---------------------------------------