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5.2. Linear Algebra (Mandatory)

5.2.1. Justification ↑ Back to top

Linear algebra is a fundamental mathematical tool shared by computer science and engineering disciplines, providing the foundations for structural analysis, circuit networks, control systems, data processing, and algorithm design. This course provides a solid foundation in the concepts and techniques of elementary and advanced linear algebra, with emphasis on its application in scientific and engineering contexts.

5.2.2. Generales Goals ↑ Back to top

1. Understand the fundamental concepts of linear algebra, including vector spaces, linear transformations, and matrix theory.
2. Apply linear algebra techniques to solve systems of equations and eigenvalue/eigenvector problems arising from scientific and engineering contexts.
3. Develop skills in abstract reasoning and logical thinking to address mathematical problems.

5.2.3. Contribution to Outcomes ↑ Back to top

AG-C07) Computing Knowledge: Applies knowledge of mathematics, science, and computing. (Familiarity)

5.2.4. Content ↑ Back to top

5.2.4.1. Matrices and Systems of Linear Equations (19 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Strang, 2016; Lay et al., 2016)

Topics

1. Matrix algebra: addition, scalar multiplication, matrix multiplication, transpose, types of matrices, and the inverse of a matrix
2. Solving systems of linear equations via Gaussian elimination and Gauss-Jordan; row echelon form, rank, and consistency of systems
3. Determinants: definition, properties, Cramer's rule, and geometric interpretation
4. Matrix decompositions: LU, QR, and singular value decomposition (SVD)

Learning Outcomes

1. Identify types of matrices and perform basic matrix algebra operations, including computing the inverse of a matrix [Familiarity]
2. Solve systems of linear equations using Gaussian elimination or Gauss-Jordan, determining the rank and consistency of the system [Usage]
3. Apply Cramer's rule and the LU, QR, and SVD decompositions to solve linear systems and factor matrices [Assessment]

5.2.4.2. Vector Spaces and Linear Transformations (14 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Strang, 2016; Lay et al., 2016)

Topics

1. Vector spaces and subspaces; bases, dimension, and linear independence
2. Linear maps, kernel and image, rank-nullity theorem, and matrix representations

Learning Outcomes

1. Identify whether a set with given operations forms a vector space and determine a basis [Familiarity]
2. Determine the kernel, image, and matrix representation of a linear map, verifying the rank-nullity theorem [Usage]
3. Apply linear transformations to model and solve problems involving change of basis and composition of maps [Assessment]

5.2.4.3. Eigenvalues, Inner Product Spaces, and Quadratic Forms (15 hours) [Skills AG-C07] ↑ Back to top

Bibliography: (Strang, 2016; Lay et al., 2016)

Topics

1. Eigenvalues and eigenvectors; diagonalization and the spectral theorem
2. Inner product spaces, orthogonality, Gram-Schmidt process, and least squares
3. Quadratic forms and definiteness: classification as positive, negative, or indefinite via eigenvalues

Learning Outcomes

1. Identify the eigenvalues and eigenvectors of a matrix and determine whether it is diagonalizable [Familiarity]
2. Compute the diagonalization of matrices via the spectral theorem and apply the Gram-Schmidt process to obtain orthogonal bases [Usage]
3. Apply least squares methods and the classification of quadratic forms via eigenvalues to solve approximation and optimization problems [Assessment]

5.2.5. Bibliography ↑ Back to top

Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.

Lay, D. C., Lay, S. R., and McDonald, J. J. (2016). Linear Algebra and Its Applications. Pearson.