Linear Algebra I: Vectors and Linear Equations

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A strong foundation in mathematics is critical for success in all science and engineering disciplines. Whether you want to make a strong start to a master's degree, prepare for more advanced courses, solidify your knowledge in a professional context or simply brush up on fundamentals, this course will get you up to speed.

In many engineering master's programs, you need to be familiar with linear algebra. This course will enable you to review the relevant topics.

This course focuses on vectors (from both algebraic and geometric perspectives) and solving linear equations. It will help you refresh your knowledge, test your skills and review the relations between the many concepts in linear algebra.

The linear algebra courses within this series will offer you an overview of this branch of mathematics common to most engineering bachelor's programs. They provide enough depth to cover the linear algebra you need to succeed in your engineering master's or profession in areas such as computer graphics, systems and control, machine learning, quantum computing and more.

This is a review course
This self-contained course is modular, so you do not need to follow the entire course if you wish to focus on a particular aspect. As a review course you are expected to have previously studied or be familiar with most of the material. Hence the pace will be higher than in an introductory course.

This format is ideal for refreshing your bachelor level mathematics and letting you practice as much as you want. Through the Grasple platform, you will have access to plenty of exercises and receive intelligent, personal and immediate feedback.

This course is part of our series Mastering Mathematics for Engineers , and together with the course Linear Algebra II part of the program Mastering Linear Algebra.

What you will learn

• Apply the dot product and cross product.
• Describe lines, planes and their intersections.
• Solve systems of linear equations and describe the solution set.
• Decide whether vectors are linear dependent or not.
• Recognize linear subspaces, describe elements of linear subspaces using bases and coordinates.
• Calculate projections and orthogonal decompositions of vectors.
• Find least-square solutions of a system of linear equations and apply it to regression.

Skills you learn

• Linear Equations
• Linear Algebra
• Quantum Computing
• Machine Learning
• Basic Math

Syllabus

Week 1: Vectors

• calculating with vectors
• the dot product
• the cross product
• lines and planes

Week 2: Linear equations

• systems of equations
• solving systems of equations
• structure of the solutions set

Week 3: Linear dependence

• linear combinations
• linear dependence
• relations between concepts

Week 4: Linear subspaces

• What are linear subspaces?
• basis and coordinates
• dimension
• the rank theorem

Week 5: Orthogonality

• orthogonal sets
• orthogonal projections
• the Gram-Schmidt algorithm
• orthogonal complements
• transposition

Week 6: Least square solutions

• "solving” an inconsistent system
• normal equations
• application to regression