Introduction to Linear Algebra: Vectors

Name: Introduction to Linear Algebra: Vectors
Rating: 4.8 (43 reviews)

Learn the basic arithmetic operations on vectors

Created byGina Chou

Last updated 1/2025

English

What you'll learn

Add and scale vectors specified geometrically or component-wise
Write the definition of a linear combination of two vectors
Write a vector as a linear combination of other vectors
Identify in simple cases if a vector is in the span of others
Write the definition of the dot product of two vectors written in components as well as the geometric definition of the dot product involving lengths and angles
Compute dot products and lengths
Find the angle between two vectors
Define and produce unit vectors
Define orthogonality
Demonstrate understanding of and apply Cauchy-Schwarz Inequality and Triangle Inequality

Course content

3 sections • 19 lectures • 1h 33m total length

Course Outline0:31
Welcome and How It Works0:30
Tips to Maximize Your Learning0:28

Downloadable Notes0:03
Geometric Vectors8:34
Explore geometric and algebraic vectors as arrows that show direction and magnitude. Learn vector addition with parallelograms and tip-to-tail methods, understand negative and zero vectors, and perform scalar multiplication.
Remarks on Geometric Vectors6:59
Example: Application of Geometric Vectors4:17
Apply geometric vectors to show that the diagonals of a parallelogram intersect at their midpoint, using midpoints and vector relations.
Algebraic Vectors9:37
Explore vector operations in algebraic form, expressing vectors by their terminal points and components in R2 and R3. Use addition, subtraction, and scalar multiplication component-wise, and note the zero vector.
Examples: Algebraic Vectors7:29
Compute multiples and linear combinations of vectors, such as 3u minus 2v, and determine parallelism by component ratios; interpret A to B as B minus A and find the midpoint.
Linear Combination5:26
Explore linear combinations of vectors as sums of scalar multiples, and express any vector as a combination of standard basis vectors i-hat and j-hat, with examples.
Span of a Set of Vectors6:10
The Norm4:15
Theorems Related to the Norm7:57
Learn the norm concepts: identify unit vectors with length one, apply nonnegativity and zero vector equivalence, and use scalar multiples to determine vector magnitudes and directions with examples.
The Dot Product6:58
Explore the dot product of vectors by relating magnitude and angle, derive u·v = |u||v|cos theta via the law of cosines, and express it through vector components.
Remarks and Examples of Dot Product8:13
Explore the dot product definition, its link to magnitudes and angle, zero means perpendicular, and cos theta via (u·v)/(|u||v|).
Theorems Related to the Dot Product7:32
Study dot product theorems: commutativity, zero vector behavior, and v·v = ||v||^2, plus distributivity and the expansion (u+v)·(u+v) = ||u||^2 + 2u·v + ||v||^2.
Cauchy-Schwarz Inequality and Triangle Inequality7:50
Explore the Cauchy-Schwarz inequality and the triangle inequality for vectors, using magnitudes, unit vectors, and the angle between vectors to bound their relationship.
Vectors

Requirements

Precalculus (algebra, trigonometry, and functions)

Description

HOW THIS COURSE WORK:

This course, Introduction to Linear Algebra: Vectors, includes the first section you will learn in Linear Algebra, including video, notes from whiteboard during lectures, and practice problems (with solutions!). I also show every single step in examples and proofs. The course is organized into the following topics:

Geometric Vectors
Algebraic Vectors
Linear Combination
Span of a Set of Vectors
The Norm
The Dot Product
Cauchy-Schwarz Inequality
Triangle Inequality

CONTENT YOU WILL GET INSIDE EACH SECTION:

Videos: I start each topic by introducing and explaining the concept. I share all my solving-problem techniques using examples. I show a variety of math issue you may encounter in class and make sure you can solve any problem by yourself.

Notes: In each section, you will find my notes as downloadable resource that I wrote during lectures. So you can review the notes even when you don't have internet access (but I encourage you to take your own notes while taking the course!).

Assignments: After you watch me doing some examples, now it's your turn to solve the problems! Be honest and do the practice problems before you check the solutions! If you pass, great! If not, you can review the videos and notes again.

THINGS THAT ARE INCLUDED IN THE COURSE:

An instructor who truly cares about your success
Lifetime access to Introduction to Linear Algebra: Vectors

HIGHLIGHTS:

#1: Downloadable lectures so you can watch the videos whenever and wherever you are.

#2: Downloadable lecture notes so you can review the lectures without having a device to watch/listen.

#3: One problem set at the end of the course (with solutions!) for you to do more practice.

#4: Step-by-step guide to help you solve problems.

See you inside the course!

- Gina :)

Who this course is for:

Anyone who has completed precalculus (algebra and trigonometry) and wants to learn some more advanced math
Current Linear Algebra students who are looking for extra help on vectors
Anyone who is not in science stream but wants to study Linear Algebra for fun

Introduction to Linear Algebra: Vectors

What you'll learn

Explore related topics

Course content

Introduction3 lectures • 1min

Vectors14 lectures • 1hr 31min

Conclusion2 lectures • 1min

Requirements

Description

Who this course is for: