Overview
On this page, we discuss how we can generalize the concept of a “vector”, but still do many of the same operations we are used to from vectors in \(\mathbb{R}^n\).
Important
The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to “read ahead” before each class  but can equally be used in a more traditional course setting.
Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch  you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
 Determine, for a given set with two operations, whether a given vector space axiom holds.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
 Determine whether a given set with two operations is a vector space: either by verifying all vector space axioms, or by showing that at least one of the axioms fails.
To prepare for class

Watch this great video (by 3Blue1Brown) which introduces the concept of a “general” or “abstract” vector space (the video also already gives an example of general linear transformations, which we will come back to later):

Watch this great video (by Socratica) which talks a bit more in detail about the vector space axioms, and shows some of the most important “abstract” vector spaces:

Watch this video (by patrickJMT) which quickly goes through the list of vector space axioms and then presents 3 example spaces which may or may not be vector spaces, for you to verify:
You should try to go through these examples yourself, but if you’re stuck or you want to verify your answer, the detailed solutions for the first two problems are here and here (he did not make a video for the third problem).