Oxford Linear Algebra: What is a Vector Space?

University of Oxford mathematician Dr Tom Crawford explains the vector space axioms with concrete examples. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: www.proprep.uk/info/TOM-Crawford
Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: www.proprep.uk/Academic/Downl...
And here: www.proprep.uk/Academic/Downl...
You can also find several video lectures from ProPrep explaining the vector space R^n here: www.proprep.uk/general-module...
And further videos explaining more general vector spaces here: www.proprep.uk/general-module...
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.
Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): • Oxford Linear Algebra:...
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: • Oxford Linear Algebra:...
What is the Determinant Function: • Oxford Linear Algebra:...
The Easiest Method to Calculate Determinants: • Oxford Linear Algebra:...
Eigenvalues and Eigenvectors Explained: • Oxford Linear Algebra:...
Spectral Theorem Proof: • Oxford Linear Algebra:...
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.
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac.uk/people/tom-c...
For more maths content check out Tom's website tomrocksmaths.com/
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
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[@TomRocksMaths]

Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: www.proprep.uk/info/TOM-Crawford

[@spencercharlton9370]

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!

[@nicktatum3340]

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

[@CatalinaAldape]

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

[@tusharhalder64]

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

[@TomRocksMaths]

Happy to help!

[@CommanderdMtllca]

That's a sick shirt

[@nicholasdavies6264]

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 ! 👍

[@Smoothcurveup52]

Wonderful explanation thank you sir

[@phenixorbitall3917]

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.

[@gurlalsingh3916]

Thanks sir

[@miegas4]

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?

[@adryel9458]

Yes. As an example: It could be the case that your vector space is one consisting of functions so the addition there is different from the one on the reals

[@zizou2076]

Yeah there is the usual addition that we know, and other additions that differ from a problem to another

[@MTahar-ig3gy]

Like the shirt

[@Azure4509]

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

[@fredtocher7180]

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

[@nikhilrajemankar1866]

You Should make it a hour long lecture, video

[@nic741]

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 :)

[@cecexhok9312]

ربي يعطيك ماتتمنى.

[@nocomment296]

Nice tshirt 👕

[@jenm1]

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

[@beastok]

I wanna wear that shirt :)

[@yuvarajghosh2789]

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

[@easymathematik]

Most of them are straight forward.