**MARÍA BARBERO LIÑÁN**

Department of Mathematics

Universidad Carlos III de Madrid

Area:

Linear algebra

Degree:

Bachelor's Degree in Statistics and Business

May, 2012

Lectures: 21 hours + Tutorials: 21 hours.

Total learning time (including the actual lectures): 140 hours.

**PRERREQUISITES AND RECOMMENDED PREVIOUS KNOWLEDGE**

1. Manipulate and simplify algebraic expressions.

2. Solve a system of two linear equations and two unknowns.

3. Basic knowledge of geometry and functions.

In general, the mathematics level achieved in high school and in professional colleges is enough.

**GENERAL COURSE DESCRIPTION**

Introduction to linear algebra.

#### OBJETIVES: KNOWLEDGE AND SKILLS

The student must be able to:

1. Solve and discuss systems of linear equations by Gaussian elimination.

2. Manipulate matrices and vectors (addition; multiplication, computation of inverse matrices and determinants whenever is possible).

3. Decide if a family of vectors are linearly dependent or independent.

4. Determine if a family of vectors spans a vector subspace. If so, find one of its basis.

5. Determine if a transformation is linear or not. If so, rewrite such a transformation in terms of matrices in different basis.

6. Determine if an endomorphism can be diagoanlized or not. Diagonalize endomorphisms whenever is possible.

7. Manipulate the abstract notions of scalar product and norm.

8. Obtain an orthonormal basis from a non-orthogonal basis by means of Gram-Schmidt method.

9. State and solve linear model problems by means of least square problems. Solve such problems with orthogonal projections.

10. Obtain singular value decomposition.

11. State and solve linear model problems by means of least square problems.

Moreover, the students must learn to:

1. Develop the ability to analize and summarize.

2. Model and solve problems.

3. Express and communicate mathematical reasoning in oral and written form.

**TEACHING MATERIALS**

Summary of lectures notes in slides.

Homework sheets with solutions.

Annotated bibliography based on D. Lay "Linear Algebra and its Applications", Addison Wesley, 2012.

**ASSESSMENT ACTIVITIES OR PRACTICAL ASSIGNMENTS**

In-class activities, online activities, in-class quizzes and final exam.

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