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# Linear Algebra and Analytical Geometry | LEB

 Course: Biomedical Engineering Curricular Unit (UC) Linear Algebra and Analytical Geometry Mandatory x Optional Scientific Area MAT Category

Course category: B - Basic; C - Core Engineering; E - Specialization; P - Complementary.

 Year: 1st Semester: 1st ECTS: 6 Total Hours: 164 Contact Hours T: 45 TP: 22.5 PL: S: OT:3 Professor in charge Isabel Matos

T - Lectures; TP - Theory and practice; PL - Lab Work; S - Seminar; OT - Tutorial Guidance

• Learning outcomes of the curricular unit

1. Perform calculations with matrices and determinants. Analyse and solve systems of linear equations.

2. Understand the concepts of vector space and linear transformation and be able to apply them to solve problems. Compute eigenvalues and eigenvectors and diagonalize matrices.

3. Compute  inner, cross and scalar triple products and understand their geometric interpretation.  Apply the concepts learned to the solution of problems in analytical geometry.

4. Apply the knowledge learned in the course to the solution of problems in engineering.

• Syllabus

1.Matrices. Definition and notation. Matrix operations. Echelon form and rank of a matrix. Systems of linear equation. Inverse of a matrix.

2.Determinants: definition and examples. Properties. Methods of evaluating determinants.

3. Vector spaces. Axiomatic definition and examples. Subspaces. Generating sets. Linear dependence. Basis and dimension. Change of basis.

4. Linear transformations. Definition and examples. Matrix representation of a linear transformation. Kernel and image of a linear transformation. Operations with linear transformations.

5. Eigenvalues and eigenvectors. Definition and examples. Eigenspaces. Algebraic and geometric multiplicity of an eigenvalue. Diagonalization.

6. Euclidean spaces. Inner product. Axiomatic definition and examples. Norm, distance,angle. The cross product and the scalar triple product. Geometrical applications.

7. Analytical Geometry. Analytical representation of straight lines and planes. Conics and quadrics.

• Demonstration of the syllabus coherence with the curricular unit's objectives.

The syllabus contains the usual tools required to solve linear problems (matrices, determinants and eigenvalue theory) and the basic examples where these tools are applied (solution of linear systems, linear maps and analytical geometry problems).

• Teaching methodologies (including evaluation

Teaching will comprise lectures and recitations, with a total of 30 classes during the semester corresponding to 67.5 contact hours (15 three hour lectures, 15 one and half hour recitations). The total work load for the student is 160 hours.

Continuous evaluation:

Two written examinations, T1 and T2,  each covering half of the syllabus. In order to pass the student must score at least 8 points (out of 20) in each exam  and average at least 9,5 points. One of the mid-term exams can be repeated on the date of the first final exam.

NF = (T1+T2)/2: NF>=10

Final exam evaluation:

In order to pass, a student must obtain a grade of at least 10 points (out of 20) in a final exam (EF) which can be attempted three times on different dates.

Formula for the calculation of Final Grade (NF):

NF = EF

Rounded to units. By defect, beneath five tenths, per excess, from five tenths.

• Demonstration of the coherence between the teaching methodologies and the learning outcomes

The lectures will present the mathematical concepts related to learning outcomes (1), (2) and (3). The recitations will focus on the solution of exercises with the aim of complete outcomes (1), (2), (3) and achieving learning outcome (4).

• Main Bibliography:

1. Anton, R.,   Rorres, C.,  "Álgebra Linear com Aplicações", Bookman.

2. Blyth S., Robertson, E.F.  "Basic Linear Algebra".  Springer.

3. Lay, D.C. , "Linear Algebra and its Applications", Pearson, Addison Wesley.

4. Monteiro, A. “Álgebra Linear e Geometria Analítica”, McGraw Hill

5. Steinbruch, A., Winterle, P.,  "Álgebra Linear", McGraw Hill.

6. Strang, G.,  "Linear Algebra and its Applications", HBJ Publishers.