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Updated in [March 06th, 2023]

This course provides an introduction to linear algebra with a focus on vectors. Students will learn how to add and scale vectors geometrically or component-wise, and how to write a vector as a linear combination of other vectors. They will also learn how to determine whether a vector is in the span of others in simple cases.

The course will cover the geometric definition of the dot product involving lengths and angles, as well as the definition of the dot product of two vectors written in components. Students will learn how to calculate the dot products and lengths, and how to determine the angle formed by two vectors. They will also learn how to define and generate unit vectors.

Finally, students will demonstrate understanding of and application of Cauchy-Schwarz and Triangle Inequality. Through this course, students will gain a better understanding of linear algebra and its applications.

At the end of the course, students will be able to add and scale vectors geometrically or component-wise, write a vector as a linear combination of other vectors, determine whether a vector is in the span of others in simple cases, calculate the dot products and lengths, determine the angle formed by two vectors, define and generate unit vectors, and demonstrate understanding of and application of Cauchy-Schwarz and Triangle Inequality.

[Applications]
After completing this course, students can apply their knowledge of linear algebra to a variety of real-world problems. For example, they can use vectors to represent physical quantities such as velocity, force, and acceleration. They can also use linear combinations of vectors to represent complex shapes and objects. Additionally, they can use the dot product to calculate the angle between two vectors, and the length of a vector. Finally, they can use unit vectors to represent directions in space.

[Career Paths]
1. Data Scientist: Data Scientists are responsible for analyzing large amounts of data and using it to make predictions and develop insights. They use a variety of tools and techniques, such as linear algebra, to uncover patterns and trends in data. Data Scientists are in high demand as businesses increasingly rely on data-driven decision making.

2. Machine Learning Engineer: Machine Learning Engineers are responsible for developing and deploying machine learning models. They use linear algebra to create algorithms that can learn from data and make predictions. They also use linear algebra to optimize the performance of their models.

3. Quantitative Analyst: Quantitative Analysts use linear algebra to develop models and algorithms that can be used to make financial decisions. They use linear algebra to analyze data and develop strategies for trading and investing.

4. Robotics Engineer: Robotics Engineers use linear algebra to design and build robots. They use linear algebra to create algorithms that allow robots to move and interact with their environment. They also use linear algebra to optimize the performance of their robots.

[Education Paths]
1. Bachelor of Science in Mathematics: A Bachelor of Science in Mathematics is a degree program that focuses on the study of mathematics and its applications. This degree program typically includes courses in calculus, linear algebra, abstract algebra, probability and statistics, and numerical analysis. Students will also learn about the history of mathematics and its applications in various fields. This degree program is ideal for those who want to pursue a career in mathematics or related fields. Developing trends in this field include the use of technology to solve complex problems, the use of data science to analyze large datasets, and the use of machine learning to automate processes.

2. Master of Science in Applied Mathematics: A Master of Science in Applied Mathematics is a degree program that focuses on the application of mathematics to solve real-world problems. This degree program typically includes courses in calculus, linear algebra, numerical analysis, optimization, and probability and statistics. Students will also learn about the application of mathematics in various fields such as engineering, finance, and economics. This degree program is ideal for those who want to pursue a career in applied mathematics or related fields. Developing trends in this field include the use of artificial intelligence to solve complex problems, the use of data science to analyze large datasets, and the use of machine learning to automate processes.

3. Doctor of Philosophy in Mathematics: A Doctor of Philosophy in Mathematics is a degree program that focuses on the advanced study of mathematics and its applications. This degree program typically includes courses in calculus, linear algebra, abstract algebra, probability and statistics, and numerical analysis. Students will also learn about the history of mathematics and its applications in various fields. This degree program is ideal for those who want to pursue a career in mathematics or related fields. Developing trends in this field include the use of technology to solve complex problems, the use of data science to analyze large datasets, and the use of machine learning to automate processes.

4. Master of Science in Data Science: A Master of Science in Data Science is a degree program that focuses on the application of mathematics and computer science to analyze large datasets. This degree program typically includes courses in calculus, linear algebra, probability and statistics, machine learning, and data mining. Students will also learn about the application of data science in various fields such as finance, healthcare, and marketing. This degree program is ideal for those who want to pursue a career in data science or related fields. Developing trends in this field include the use of artificial intelligence to solve complex problems, the use of natural language processing to analyze text data, and the use of deep learning to automate processes.

Course Syllabus

Geometric Vectors

Remarks on Geometric Vectors

Example: Application of Geometric Vectors

Algebraic Vectors

Examples: Algebraic Vectors

Linear Combination

Span of a Set of Vectors

The Norm

Theorems Related to the Norm

The Dot Product

Remarks and Examples of Dot Product

Theorems Related to the Dot Product

Cauchy-Schwarz Inequality and Triangle Inequality