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Clase 1 - El método de eliminación de Gauss

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Capítulo 1.2 del texto guía (Grosman & Flores, 7a edición):

P. 17: Ejemplo 1.2.7 (inténtenlo primero sin mirar la solución).

P. 23:Auto-evaluación 1.2.

PP. 26-27: Problemas 1-27 (hagan unos cuantos, hasta sentirse seguros de que han entendido).

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Vectors | Chapter 1, Essence of linear algebra

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Capítulo 2.1:

PP.55-56: Resumen y auto-evaluación 2.1.

PP. 58-60: Problemas 1-52.

Capítulo 2.2:

PP. 77-78: Resumen y auto-evaluación 2.2.

P. 81: Problemas 20-36.

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Herramienta de cálculo científico en línea (Sage, Python, R)

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¡Lo recomiendo muchísimo!

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Powerful Compute Servers with Optional GPU Support

Extend your CoCalc projects with powerful compute servers. They give you much more power, optional GPU support, and flexibility for your computations.

From within your CoCalc project, spin up and connect to a powerful machine. You simply tell your terminals and Jupyter Notebooks to run on these machines, or with one click launch JupyterLab or VS Code. These compute servers open up new possibilities by utilizing enhanced computing resources, extending far beyond the bounds of what you can do in your local project.

These machines optionally come with extensive GPU support, from a single NVIDIA T4 to eight H100's, with many options in between, including L4 and L40, RTX-A4/5/6000, and A100 with 40GB and 80GB. The finely configured software images include SageMath, Anaconda, Google Colab, Julia, PyTorch, Tensorflow and Open WebUI, accommodating a versatile range of uses. The pre-configured software environments make it very easy to make use of them, right out of the box. You can also run any command as root, install anything you want, and use Docker and Kubernetes.

Your files are synchronized . Therefore, you can seamlessly switch between local and remote computing. You also have much more disk storage on the remote machine.

Usage of these machines is billed by the second. The pricing is highly competitive, starting at under $0.01/hour and under $0.15/hour with a GPU!

More about compute servers on CoCalc

Extensive Generative AI Integration

Many Programming Languages

CoCalc supports many programming languages. Edit the demo cell on the left and evaluate it by pressing "Run". You can also select a different "kernel", i.e. the programming language that is used to evaluate the cell.

Realtime Collaboration Using Your Favorite Tools

Everyone's code runs in the same per-project environment, which provides consistent results, synchronized file changes, and automatic revision history so that you can go back in time when you need to discover what changed and when. You can even publish your CoCalc creations to share with anyone via the built-in share server.

Forget the frustration of sending files back and forth between your collaborators, wasting time reviewing changes, and merging documents. Get started with CoCalc today.

Jupyter, SageMath, LateX, and Linux

CoCalc's own Jupyter Notebook implementation offers realtime synchronization, TimeTravel, automatic grading, side chat, and more.

Sage Worksheets are similar to Jupyter Notebooks, but made to work well with SageMath. They offer a single-document model that scales to large documents and integrated 3d graphics.

The very sophisticated collaborative Linux Terminal makes you incredibly productive. Many programming languages and hundreds of tools are available at your fingertips in a full Ubuntu Linux environment.

Integrated Course Management System

You can think of CoCalc as virtual computer lab in the cloud. It takes away the pain of teaching scientific software.

Hassle-free assignments: CoCalc keeps all files well organized! Due to real-time synchronization you never have to deal with multiple versions of the same file. There is even support for automated grading via NBGrader.

Pre-installed Software like in a computer lab, all software you need is already installed and ready to use.

Real-time Collaboration allows you to virtually look students over their shoulders. You can check their work directly and help them by editing the file or using side-chat communication.

More about teaching on CoCalc

Much More ...

Use a full featured collaborative whiteboard - with support for Jupyter notebook cells - to express and share your ideas.

CoCalc offers a variety of features to make your life easier. You can find a list of all features here.

Solutions

Upgrade your projects to unlock internet access, better hosting quality, and other upgrades by purchasing a license or upgrade via pay-as-you-go and use GPUs and HPC resources via compute servers!

The course license options are very flexible: they range from small professional training up to large university courses. The students can pay CoCalc directly, or you can pay on their behalf, and it is easy to change a license at any time if you need more resources or the number of students changes.

It is very easy to run CoCalc on your own computer or cluster.

There are three options available:

1. Make your computer available in a CoCalc project via an on-prem compute server.

2. Deploy a highly scalable variant of CoCalc on your Kubernetes cluster via cocalc-cloud.

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Collaborative Calculation and Data Science

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También existe en español.

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Matrix addition, multiplication, inversion, determinant and rank calculation, transposing, bringing to diagonal, row echelon form, exponentiation, LU Decomposition, QR-decomposition, Singular Value Decomposition (SVD), solving of systems of linear equations with solution steps

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Clase 3 - Matrices y resolución de sistemas

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Combinaciones lineales, espacio generado y bases

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Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra

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Capítulo 2.3:

Matrices aumentadas y resolución de sistemas: Ejercicios 21 a 29 p.99.

Para ir más lejos (opcional): Ejercicios 30, 31, 32 p.100.

Capítulo 2.6:

Ejercicios 47 a 73 pp. 143-144.

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Linear transformations and matrices | Chapter 3, Essence of linear algebra

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La multiplicación matricial vista como composición de transformaciones lineales

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Matrix multiplication as composition | Chapter 4, Essence of linear algebra

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Capítulo 2.4:

Determinar si una matriz es invertible y calcular su inversa: Ejercicios 17 a 22 p.118.

Para ir más lejos (opcional): Ejercicio 23 p.118 y Ejercicios 30 a 37 p.119.

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Matrices no cuadradas vistas como aplicaciones de un espacio a otro

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Nonsquare matrices as transformations between dimensions | Chapter 8, Essence of linear algebra

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The determinant | Chapter 6, Essence of linear algebra

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Capítulo 3.1:

Ejercicios 1 a 11 p.187.

Para ir más lejos: Ejercicios 17 a 20 p.187 (el ejercicio 18 contiene un error: no es ”AB = det(A) det(B)” sino "det(AB) = det(A) det(B)”.

Capítulo 3.2:

Auto-evaluación p.205.

Ejercicios 28 a 36 pp.206-207.

Ejercicios 41, 42 p.207.

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Matrices inversas, espacio de columnas y espacio nulo

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Inverse matrices, column space and null space | Chapter 7, Essence of linear algebra

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Clase 6 - Determinantes y matrices invertibles

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Clase 6 - Determinantes y matrices invertibles

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Interpretación geométrica de la regla de Cramer

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Cramer's rule, explained geometrically | Chapter 12, Essence of linear algebra

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Capítulo 3.3:

Auto-evaluación pp.215-216.

Ejercicios 23 y 24 p.217.

Capítulo 3.5:

Ejercicios 1 a 19 pp.229-230 (ejercicios de repaso).

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Clase 9 - De sub-espacios a espacios vectoriales

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Capítulo 5.1:

Auto-evaluación 5.1 p.302 del libro (pueden quitar los ítems (IV) y (V)).

Ejercicio 32 p.304.

Capítulo 5.2:

Auto-evaluación 5.2 p.312 del libro.

Ejercicios 22 a 37 p.313.

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Abstract vector spaces | Chapter 16, Essence of linear algebra

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Capítulo 5.3:

Auto-evaluación 5.3 p.319.

Problemas 1 a 25 p. 320-321 (algunos, no necesariamente todos).

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Capítulo 5.4 (Independencia lineal / Familias libres):

Ejemplos 5.4.8 p.339 y 5.4.10 p.340.

Auto-evaluación 5.4 p.341.

Para ir más lejos: Ejercicios 26,27 p.341.

Capítulo 5.5:

Problemas 1 a 14 p.358.

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Capítulo 5.5:

Auto-evaluación 5.5 p.357.

Ejercicios 33, 34 p.359.

Para ir más lejos: Ejercicio 35 p.359 (difícil).

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Capítulo 5.6:

Ejemplos 5.6.1, 5.6.2, 5.6.3 y 5.6.4 pp.366-369 del libro.

Auto-evaluación 5.6 p.372.

Problemas 22, 23 y 21, 24, 25 pp.374-375.

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Change of basis | Chapter 13, Essence of linear algebra

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Clase 14 - Dimensión del espacio nulo, del espacio de columnas, y aplicaciones geométricas

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Capítulo 5.7:

Auto-evaluación 5.7 p.396: Preguntas I a VII únicamente.

Problemas 5, 7, 8, 12, 15, 18, 21 p.398-399.

Problemas 22 a 28 y 29 a 33 p.399.

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Clase 17 - Transformaciones lineales entre espacios vectoriales

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Capítulo 7.1:

Ejemplos 7.1.3 p.482, y 7.1.7, 7.1.8 p.483.

Auto-evaluación 7.1 p.486.

Para ir más lejos: Problema 52 p.488.

Capítulo 7.2:

Auto-evaluación 7.2 p.499.

Problemas 1 a 9 p.500.

Para ir más lejos: Problemas 25 y 26 p.501.

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Capítulo 7.3:

Los ejercicios 18.7, 18.11 y 18.12 de las notas del curso.

Ejemplo 7.3.9 p.509 del libro.

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Clase 19 - Cambio de base, matrices semejantes

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Capítulo 7.3:

Auto-evaluación 7.3 p.518.

Problemas 1 a 8 p.518.

Problemas 17 a 19 y 26,27 p.519.

Ejercicio 6 p.525 (con SageMath, Python o R para los cálculos si quieren).

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Clase 20 - Aplicaciones lineales invertibles

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Capítulo 7.4:

Auto-evaluación 7.4 p.532.

Problemas 7, 8, 9 p.533.

Problema 14 p.533.

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Clase 21 - Valores propios y vectores propios

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Clase 21 - Valores propios y vectores propios

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Capítulo 8.1:

Auto-evaluación 8.1 p.559: Preguntas I, II, III y V.

En los Problemas 2, 5, 6 y 9 pp.560-561, hallen los valores propios de la matriz propuesta y determine la dimensión de los sub-espacios propios de esa matriz.

Misma pregunta para los Problemas 15, 17 y 20 p.561.

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Clase 22 - Diagonalización y aplicaciones (se podría agregar trigonalización, cf Grifone p.218)

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Capítulo 8.3:

Ejemplos 8.3.4 p.582 y 8.3.7 p.584.

Ejemplos 8.3.5 y 8.3.6 p.583.

Auto-evaluación 8.3 p.586.

Problemas 1 a 15 p. 588.

Problemas 27, 28 y 30 p.589.