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1. Systems of linear equations. Equivalent systems of linear equations. Gauss method and classification of a system with respect to the number of solutions.                                                                                   
2. Matrices:  Square matrices, diagonal, upper and lower triangular matrices. Addition of matrices and product of a matrix by a scalar. The identity matrix. The rank of a matrix. Matrix form of a linear system and classification of the system using the rank of the matrix. Invertible (square) matrices. Uniqness of the inverse. The inverse of the product of two invertible matrices. The rank of an invertible matrix. How to compute the inverse of a matrix.The transpose of a matrix and the transpose of the product of two matrices.   
       

3. Determinants. Definition, properties and examples. Relation between the determinant and elementary operations on the rows of the matrix. Relation between the rank of a matrix being maximal and the determinant being non-zero. Laplace theorem and Sarrus Rule. The adjoint matrix, its determinant and how to compute the inverse of a matrix using the adjoint. Cramer systems and their solutions. Cross product and its geometrical interpretation. The absolute value of the determinant of a 2x2 matrix resp. a 3x3 matrix.                                      

4. Vector spaces. Definition, properties and examples. Vector subspaces, description of the ones of R2 and of R3. 

Sum and intersection of vector subespaces. The vector space of solutions of an homogeneous linear system. Construction of solutions of a linear system using the solutions of the associated homogeneous linear system. Linear combination. Subspace generated by a set of vectors. Linear and independence linear. Steinitz exchange lemma. The concept of basis and dimension of a vector space. Relation between the dimension of the subspace generated by vectors in Rn and the rank of the matrix formed by these vectors. Line and column rank. Ordered basis and coordinates with respect to a basis. Invertibility of a change of basis matrix. Direct sums of subspaces and its dimension. Complementary subspaces.

Subspaces as solutions of a system of linear equations. Revisions about lines and planes through the origin in R2 and in R3.

5. Linear maps. Definition and examples. The image and the inverse image of a subspace by a linear map. Defining a linear map using only the images of the elements of o basis. Matrix of a linear map. Relation between two matrices of the same linear map with respect to different basis. The composition of linear matrices and its matrix. Isomorphisms. Description of the matrix of the inverse of an isomorphism. The kernel of a linear map. The Kernel of a monomorphism. The theorem of dimensions. Indentification, up to isomorphim, of a finite dimensional vector space over K with Kn. Dimension of the vector space of all linear maps between finite dimensional spaces. Identificantion between matrices and linear maps between finite dimensional vector spaces.      Determinant of an endomorphism of a finite dimensional vector space.