--- title: Systems of Linear Equations description: Systems of linear equations are collections of linear equations sharing common variables. You will understand how to represent these systems using augmented matrices and solve them systematically using Gaussian elimination, back-substitution, and matrix inversion techniques. category: mathematics subcategory: linear-algebra difficulty: beginner, intermediate, advanced url: /subject/systems-of-linear-equations --- # Systems of Linear Equations Systems of linear equations are collections of linear equations sharing common variables. You will understand how to represent these systems using augmented matrices and solve them systematically using Gaussian elimination, back-substitution, and matrix inversion techniques. ## Available Resources 2 Videos • 3 Books • 3 Courses • 2 Websites ## Videos ### 1. Introduction to Linear Algebra: Systems of Linear Equations **Author:** Dave Farina A short lecture from Professor Dave Explains introducing linear algebra through systems of linear equations. It shows how such systems are written and solved and how they lead to matrices, giving a first footing before studying matrix methods. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/watch?v=csgNflj69-Y **Tags:** linear-algebra, linear-equations, systems-of-equations, matrices ### 2. Vectors | Chapter 1, Essence of Linear Algebra **Author:** Grant Sanderson Opening chapter of Grant Sanderson's animated linear algebra series, contrasting the physics, computer science and mathematics views of vectors. Shows vectors as arrows and coordinate lists, and how addition and scalar multiplication work geometrically, laying intuition for linear combinations and transformations. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/watch?v=fNk_zzaMoSs **Tags:** linear-algebra, vectors, vector-addition, visual-intuition ## Courses ### 1. Linear Algebra (MIT 18.06) **Author:** Gilbert Strang Matrix theory and linear algebra: systems of equations, elimination, vector spaces and subspaces, orthogonality and least squares, determinants, eigenvalues and positive definite matrices. Includes 35 lecture videos plus problem sets and exams with solutions, giving the fluency needed for applied mathematics, engineering and data science. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/ **Tags:** linear-algebra, vector-spaces, eigenvalues, least-squares, matrix-factorization ### 2. Linear Algebra - Foundations to Frontiers Learn the mathematics behind linear algebra and link it to matrix software development. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.edx.org/learn/linear-algebra/the-university-of-texas-at-austin-linear-algebra-foundations-to-frontiers **Tags:** courses, mathematics-statistics, linear-algebra ### 3. Mathematics for Machine Learning: Linear Algebra In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. Then we look through what vectors and matrices are and how to work with them, including the knotty problem of eigenvalues and eigenvectors, and how to use these to solve problems. Finally we look at how to use these to do fun things with datasets - like how to rotate images of faces and how to extract eigenvectors to look at how the Pagerank algorithm works. Since we're aiming at data-driven applications, we'll be implementing some of these ideas in code, not just on pencil and paper. Towards the end of the course, you'll write code blocks and encounter Jupyter notebooks in Python, but don't worry, these will be quite short, focussed on the concepts, and will guide you through if you’ve not coded before. At the end of this course you will have an intuitive understanding of vectors and matrices that will help you bridge the gap into linear algebra problems, and how to apply these concepts to machine learning. **Difficulty:** Beginner | **Language:** English | **Duration:** 5 weeks of study, 2-5 hours/week | **Price:** Free **Link:** https://www.coursera.org/learn/linear-algebra-machine-learning **Tags:** courses, mathematics-statistics, linear-algebra ## Books ### 1. Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares (free PDF) **Author:** Stephen Boyd, Lieven Vandenberghe Stanford and UCLA authors' applied linear algebra text, free PDF from Cambridge. Covers solving square and overdetermined systems via QR factorization and back substitution, with operation counts, so you understand what a solver actually does and when least squares replaces exact solution. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://web.stanford.edu/~boyd/vmls/ **Tags:** linear-algebra, least-squares, qr-factorization, matrices, applied-math ### 2. Introduction to Linear Algebra, 6th Edition **Author:** Gilbert Strang The textbook behind MIT 18.06, now in its sixth edition. It builds elimination, augmented matrices, and LU factorization into the four-subspaces framework, so you can read off whether Ax=b has one solution, infinitely many, or none. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://math.mit.edu/~gs/linearalgebra/ila6/indexila6.html **Tags:** linear-algebra, gaussian-elimination, lu-factorization, four-subspaces, systems-of-equations ### 3. Linear Algebra and Its Applications **Author:** David C. Lay Undergraduate linear algebra textbook that builds from row reduction of linear systems through matrix algebra, determinants, vector spaces, eigenvalues, orthogonality, and least squares, with applications from economics, engineering, and computer graphics. Readers learn to solve and analyze systems of equations using matrices. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0201824787?tag=edmonddante07-20 **Tags:** linear-algebra, matrices, linear-systems, eigenvalues, vector-spaces, least-squares ## Websites ### 1. Interactive Linear Algebra (Georgia Tech) **Author:** Dan Margalit, Joseph Rabinoff Free online textbook from Georgia Tech's Math 1553 with embedded 3D demos you can drag. Two full chapters treat linear systems algebraically and geometrically, so you see row reduction, consistency, and solution sets as intersecting planes and spans. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://textbooks.math.gatech.edu/ila/ **Tags:** linear-algebra, systems-of-equations, row-reduction, geometric-intuition, interactive-textbook ### 2. mathsisfun.com MathIsFun.com is a beginner-friendly math site that explains concepts with clear, step-by-step lessons and interactive activities. It covers topics from algebra and linear equations to geometry, fractions, data, and more, with practice problems, worksheets, and quizzes. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathsisfun.com **Tags:** websites, mathematics-statistics, algebra ## Youtubes ### 1. Inverse Matrices, Column Space and Null Space — Essence of Linear Algebra Ch. 7 **Author:** Grant Sanderson Fifteen-minute animated lesson recasting Ax=v as a geometric question. You will understand why a nonzero determinant guarantees a unique solution, what column space and rank say about solvability, and how the null space describes the full solution set. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.youtube.com/watch?v=uQhTuRlWMxw **Tags:** linear-algebra, inverse-matrices, column-space, null-space, rank --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/systems-of-linear-equations