Oxford Linear Algebra: What is a Vector Space?

As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.

Watch other videos from the Oxford Linear Algebra series at the links below.

The video begins by introducing the vector space axioms. We first define the addition and scalar multiplication maps, before listing the 8 axioms that must be satisfied: commutativity of addition, associativity of addition, the existence of an identity element, the existence of additive inverses, distributivity of scalar multiplication over addition, distributivity of scalar multiplication over field addition, interaction of scalar multiplication and field multiplication, and the existence of an identity for scalar multiplication.

Each axiom is then verified for 3D coordinate vectors as a canonical example. Finally, further properties of vector spaces are discussed, such as the uniqueness of identity elements and inverses. A full proof using the axioms is provided to show the additive identity is unique.

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[spencercharlton]

Really great video! I just wrapped up the intro linear algebra course of my undergrad last semester. I'm happy to see such a great video on one of the topics I struggled most with. Once all of the abstractness became clear I realized there are so many applications for vector spaces and I think its so cool. I plan to study linear algebra in future years of university. Sending love from Canada! The content is awesome! <3

[nicktatum]

Love and support from a Math/Numbers fan here in the US✊🏼❤️💪🏽 #️⃣♾

[tusharhalder]

You are amazing Prof. Tom! I’ve learnt a lot from your vids and it really did help me in my grades.

[nicholasdavies]

Hi Tom, could you do a vlog relating to your University standard mathematics and how it relates to industry or other ?
If you gain a degree in Mathematics... where does the end result lie, work wise ?
Enjoy your vlogs ! 👍

[nic]

I had a quick question which might sound silly, but I was hoping you could help resolve. As I understand, when you say that a space (i.e: Vector space) is "equipped", then this means that this space has a map built into it. However, my confusion is about the vector space axioms. Are these axioms a result/consequences of these maps or are they something extra we impose on the space to make it into a vector space? Thank you in advance :)

[phenixorbitall]

Nice! Thank you very much. Could you please also make a video on Galois theory one day Sir? I really like the way you explain things.

[CatalinaAldape]

Thanks for posting this! Such a helpful breakdown. :)

[miegas]

In axiom vi) it seems that this axiom is stating the relationship between the + operator of the vector space and the addition of the field. they seem to my untrained eye to be 2 different operations using the same symbol. Is this what is going on?

[Azure]

Can you please state the difference between Space and Structure in math?

[jenm]

I assume this is past first semester Lin Alg bc I hardly know what's going on

[fredtocher]

Don’t even study maths just like the way tom explains stuff

[nikhilrajemankar]

You Should make it a hour long lecture, video

[yuvarajghosh]

Sir please try to solve the Maths section of India's JEE Advanced papers. And give a review in your channel.