Enroll Course: https://www.coursera.org/learn/introduction-to-linear-algebra
Linear algebra is a cornerstone of modern mathematics, underpinning advancements in fields ranging from computer science and engineering to biology and economics. If you’re looking to build a solid understanding of this crucial subject, Coursera’s ‘Introduction to Linear Algebra’ is an excellent starting point.
This course, designed to be taken in parallel with calculus, highlights the inseparable nature of these two mathematical pillars. The overview emphasizes how linear algebra, alongside calculus, forms the backbone of countless applications in the physical and biological sciences, engineering, and computer science. It’s not just about abstract theory; it’s about understanding the tools that drive innovation.
The syllabus is structured logically, guiding learners through essential concepts:
**Week 1: Geometric Vectors in the Plane and in Space**
This introductory module dives into the world of geometric vectors. You’ll learn to treat vectors as directed line segments, mastering their arithmetic in both 2D and 3D space. This practical approach helps build intuition for vector quantities ubiquitous in the physical world. The skills gained here are foundational for understanding more abstract vector space theories and preparing for advanced linear algebra topics.
**Week 2: Lines and Planes in Space and Systems of Linear Equations**
Building on vector arithmetic, this week focuses on representing and manipulating lines and planes in space using vector and Cartesian equations. You’ll tackle optimization problems, like finding closest points between geometric objects, and delve into systems of linear equations. The course introduces Gaussian elimination, elementary row operations, and back substitution for solving these systems, even exploring solutions within modular arithmetic.
**Week 3: Matrix Arithmetic and the Theory of Determinants**
Matrices are introduced as a powerful tool for organizing and manipulating data. You’ll learn the fundamentals of matrix addition, scalar multiplication, and matrix multiplication. The module covers elementary matrices, matrix inverses, and the crucial concept of determinants, including their properties and their role in identifying invertible matrices. Applications to vector cross products and spatial relationships are also explored.
**Week 4: Eigentheory and Diagonalisation**
The final week delves into eigenvalues and eigenvectors, which reveal the fundamental directions of transformation under a linear operator. This leads to the powerful technique of diagonalisation, useful for solving complex matrix problems, such as calculating powers of matrices or analyzing Markov processes. You’ll also explore linear transformations, including rotations and reflections, and get a glimpse into the proof of the Fundamental Theorem of Algebra, solidifying the understanding that all square matrices possess eigenvalues.
**Recommendation:**
Coursera’s ‘Introduction to Linear Algebra’ is a comprehensive and well-structured course. It effectively bridges the gap between theoretical concepts and practical applications, making linear algebra accessible and engaging. The progression through geometric vectors, systems of equations, matrices, and finally eigenvalues, provides a robust foundation for anyone looking to deepen their mathematical understanding or apply these concepts in STEM fields. Whether you’re a university student needing to supplement your coursework or a professional looking to upskill, this course is highly recommended.
Enroll Course: https://www.coursera.org/learn/introduction-to-linear-algebra