What is a Vector Space? (Abstract Algebra)

Vector spaces are one of the fundamental objects you study in abstract algebra. They are a significant generalization of the 2- and 3-dimensional vectors you study in science. In this lesson we talk about the definition of a vector space and give a few surprising examples.

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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition

Milne, Algebra Course Notes (available free online)

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Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro
Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison
Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison

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[Rainsong]

Thank you so much for making it easy and simple to understand, and yet not "talking down" to any new students of the subject.

[alanveiga]

You explained in 7 minutes what my professor couldn't explain in a whole semester during two hours lectures.

[rationaloptimist]

One of the few who don't give a text book definition but actually try to explain the logic behind it.

[WithASideOfFries]

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I like that socratica chooses to cover abstract subjects.

[Songvbm]

I swear to god, I am in love with this channel. Linear Algebra is NOW MADE EASY.

[danielsmirnov]

Arguably the best and simplest explanation of a vector space I've ever heard. Thanks a bunch!

[ARCTCH]

Thank I'm taking linear algebra with a midterm test in 2 days. The book throws all these rules about vector spaces but never does it take the time to explain or help visualize WHAT a vector space actually represents. I can memorize the rules but I just felt so lost.... But this really helped me.

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[arya]

You are the maths teacher I have been looking for. Thank you so much socratica for helping many of us to imagine the concepts from maths.

[kaycee]

I have an exam in three days, and you just helped me understand vector space better than my teacher and textbook. I feel more comfortable with the topic and have a better grasp of it. Thank you!

[chandrasekarnarayan]

Vectors addition, multiplication rules can be a applied to n-dimensions. In Computer Games like Age of Empires 5 dimensions are Food, Wood, Gold, Stone, Population.

[cristian-bull]

A physicist and a mathematician are in a congress and the talk goes like:
"If we consider an 8 dimensional sphere, we can show that the radio..."
The physicist, skeptical, ask the mathematician:
- "Do you believe such nonsense about 8 dimensional spaces?"
- "Sure, of course"
- "So.. how do you "see" an 8 dimensional sphere?"
- "Easy. I imagine a N dimensional sphere, and then set N equals 8".

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This is awesome stuff, Socratica!!! It's very, very easy to become a huge fan of abstract algebra watching videos like these!!! :) :)

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[mukulverma]

I'm speechless. Such a clear and precise explanation, really makes me happy that I stumbled upon your channel! Big thanks from a student who was facing a hard time understanding these abstract concepts. I'm going to watch all the videos related to my syllabus, that has been covered by you guys. Peace!

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