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Importance of Linear Algebra

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Importancia del Álgebra Lineal en la Vida Diaria
Adribel González
20143098

El álgebra lineal es vital en múltiples áreas de la ciencia en general. Debido a que las ecuaciones lineales son tan fáciles de resolver, prácticamente todas las áreas de la ciencia moderna contiene modelos en los que las ecuaciones se aproximan mediante ecuaciones lineales y resolviendo el sistema ayuda a desarrollar la teoría. Dado que en la mayoría de los casos, la resolución de ecuaciones es un sinónimo de resolver un problema práctico, esto puede ser muy útil. Sólo por esta razón, el álgebra lineal tiene una razón de ser, y es una razón suficiente para cualquier estudiante aprender álgebra lineal.

La informática ha entregado extraordinarios beneficios en las últimos décadas. La amplitud y profundidad de estas aportaciones se están acelerando junto a un mundo que se conecta globalmente. Al mismo tiempo, el campo de la informática tiene como objetivo el tocar casi todas las facetas de nuestras vidas. El álgebra lineal ha jugado un papel importantísimo en estos avances, pues ha permitido que la integración entre las áreas de la matemática sea más simple y eficiente, permitiendo que problemas muy complejos puedan ser divididos e iterados con técnicas relativamente sencillas.

En el día a día, con solo encender una computadora, cruzar un puente o usar detergente estamos poniendo en uso el álgebra lineal. Las matrices se utilizan para estudiar cosas como cadenas de Markov, que tienen aplicaciones prácticamente en todas partes; por ejemplo, el algoritmo PageRank de Google, básicamente, hace los cálculos con una matriz gigante creada del Internet. Los químicos utilizan matrices para estudiar los posibles modos de vibración de los átomos y las moléculas.

Existen un sinnúmero de ejemplos, sin embargo nada más simple que las calles que transitamos a diario. El álgebra lineal es la forma más sencilla de ver las funciones de muchas variables, que por lo general se presentan en ingeniería por la discretización de un concepto establecido en términos de un continuo, por ejemplo, la ley que rige la relación entre el estrés y las tensiones en una estructura. Esta rama matemática se utiliza muy fuertemente en Ingeniería Estructural. El análisis de una estructura en equilibrio implica escribir muchas ecuaciones con muchas incógnitas. A menudo, estas ecuaciones son lineales, incluso cuando se considera la deformación del material (es decir, flexión).
Este es exactamente el tipo de situación para la que el álgebra lineal es la mejor técnica.

En mi carrera, la computación, el álgebra por lo general se puede dividir en dos categorías: El álgebra lineal para cantidades espaciales. Aquí se está tratando con 2, 3, o 4 vectores dimensionales y se está preocupado por las rotaciones, proyecciones y otras operaciones con matrices que tienen alguna interpretación espacial. Esta es la clase de álgebra lineal que aparece, por ejemplo, en los gráficos de computadora o simulaciones de física.
Y el álgebra lineal para las estadísticas. Aquí se está tratando con vectores en espacios de alta dimensión que no tienen una interpretación espacial particular y se está interesado en las descomposiciones de la matriz y así sucesivamente. Este dominio incluye el procesamiento de señales, aprendizaje automático estadístico, y compresión.

En los video juegos, otro de mis pasatiempos, el álgebra lineal tiene mucha fuerza. Los gráficos 3D tienen que ver directamente con el álgebra lineal. El trazado de formas, incluyendo la rotación de ellos, moverlos, colocando cámaras en ciertos lugares, la física

en 2D y 3D, así (fuerzas, colisiones, etc.) En las películas y los renders, los efectos 3D programados con shaders utilizan álgebra lineal y hardware para calcular grandes cantidades de cálculos muy eficientemente utilizando procesamiento paralelo.

En términos económicos, la álgebra lineal es un pilar de prácticamente toda aplicación financiera. Desde calcular amortización hasta evaluar cuál opción es más costo-eficiente, las ecuaciones lineares nos permiten realizar estas operaciones. Para una casa, por ejemplo, respondemos preguntas como "¿cuánto quiero terminar pagando por todo el plazo de la hipoteca, y cómo se compara esto con el valor de la casa?" Y "¿Qué tan grande es la diferencia en los pagos mensuales de entre una hipoteca a 15 años y una hipoteca a 30 años?“

El álgebra lineal es tan amplia en sus usos, que incluso encuentra hogar en la política, donde sus técnicas son necesarias para tomar decisiones de las cuales dependen un sinnúmero de vidas. Por ejemplo, si se construyen más carreteras, la capacidad sube lo que es bueno para los negocios, pero la calidad de vida disminuye. La gente se muda más lejos por la facilidad, lo que se traduce en más tráfico, lo que se traduce en menos exceso de capacidad, lo cual es malo para los negocios. Muchas variables y conexiones complejas conducen a un cálculo multivariante. Los sociólogos y los jefes en la política deben saber álgebra lineal para pensar y resolver en problemas como estos, pudiendo así tomar decisiones más acertadas y eficientes.

Como la mayoría de las personas nunca llegan a notar estos usos, aprender álgebra lineal nos da una visión del mundo que la mayoría de la gente no ve. Eso añade una dimensión humana interesante para nuestro desarrollo como profesionales y como seres pensantes.
Desarrollar la habilidad de poner un problema en perspectiva, nos prepara para tomar

decisiones educadas y costo-eficientes. El saber manejar complejos sistemas nos dará un entendimiento de nuestros alrededores que sólo así podremos obtener.

Es así como el álgebra lineal es útil en cualquier momento en que nos encontremos con múltiples variables, lo cual sucede mucho más de lo que pensamos. Es útil cuando se tienen muchas partes móviles de un motor, concentraciones de un producto químico en un tubo de ensayo, muchas regiones de la atmósfera en una simulación del clima, precios de las acciones de un mercado, etc .. A pesar de que algunas aplicaciones se valen de sistemas no lineales, el álgebra lineal puede y simplifica lo más complejo para nuestro uso, regalándonos así toda una gama de profesiones, avances, y descubrimientos que nos impulsan al futuro.

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