## Courses Catalogue

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# Linear Algebra and Analytic Geometry C

5240

### Department

Departamento de Matemática

6.5

### Teacher in charge

Júlia Maria Nunes Loureiro Vaz de Carvalho, Maria Cecília Perdigão Dias da Silva

5

Português

### Objectives

The student is supposed acquire basic knowledge on Linear Algebra (vide Program) and that, in learning process, logical reasoning and critical mind are developed.

### Prerequisites

The student must be familiar with mathematics taught at pre-university level in Portugal (science area).

### Subject matter

LINEAR ALGEBRA AND ANALYTIC GEOMETRY - C

1  Matrices: Definitions and basic results. Row-echelon form. Matrices and elementary row/column operations. Characterization of invertible matrices and determination of the inverse.

2  Systems of Linear Equations: Equivalent systems. Matricial representation of a system of linear equations. Resolution and discution of systems.

3  Determinants: Definition and properties. Determinant of the product. Classical adjoint (adjugate) of a matrix. Computation of the inverse from the adjugate.

4  Vector Spaces: Definition and properties. Subspaces. Intersection and sum of subspaces and the relation of their dimensions. Linear combinations and subspace generated by a system of vectors. Principal results about linear dependence/independence of a system of vectors. Bases. Extension to a basis of a linearly independent system of vectors.

5  Linear Transformations: Properties. Dimension theorem and other fundamental results. Matrix of a linear transformation and of composition of transformations. Matrices and changing of bases.

6  Eigenvalues and Eigenvectors: Eigenvalues and eigenvectors of a matrix/linear operator. Eigenspaces. Algebraic and geometric multiplicity. Diagonalisable matrices/linear operators.

7 - Inner, Vector and Mixed Products: Definitions and properties in R3.

8  Analytic Geometry: Cartesian representations of the straight line and the plane. Metric and no metric problems.

### Bibliography

MAIN REFERENCE:

ISABEL CABRAL, CECÍLIA PERDIGÃO, CARLOS SAIAGO Álgebra Linear e Geometria Analítica, 2006/2007.

OTHER REFERENCES:

J. V. DE CARVALHO Apontamentos da disciplina de Álgebra Linear e Geometria Analítica, Departamento de Matemática, Universidade Nova de Lisboa, Ano Lectivo 2000/2001.

E. GIRALDES, V. H. FERNANDES, M. P. M. SMITH Álgebra Linear e Geometria Analítica, McGraw-Hill de Portugal, (1995).

A. MONTEIRO Álgebra Linear e Geometria Analítica, McGraw-Hill de Portugal, (2001).

T. S. BLYTH, E. F. ROBERTSON Basic Linear Algebra, Springer-Verlag, (1998).

H. ANTON, C. RORRES Elementary Linear Algebra - Applications version, 8th Edition, John Wiley & Sons, Inc. (2000).

### Evaluation method

There are tree tests that can substitute the final exam in case of approval. Otherwise the student must succeed the final exam. More detailed rules are available in the portuguese version.