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: https://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: https://www.proprep.uk/Academic/Downl...

And here: https://www.proprep.uk/Academic/Downl...

You can also find several video lectures from ProPrep explaining the vector space R^n here: https://www.proprep.uk/general-module...

And further videos explaining more general vector spaces here: https://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: Elementary Row...  

Calculating the inverse of 2x2, 3x3 and 4x4 matrices:    • Oxford Linear Algebra: How to find a ...  

What is the Determinant Function:    • Oxford Linear Algebra: What is the De...  

The Easiest Method to Calculate Determinants:    • Oxford Linear Algebra: The Easiest Me...  

Eigenvalues and Eigenvectors Explained:    • Oxford Linear Algebra: Eigenvalues an...  

Spectral Theorem Proof:    • Oxford Linear Algebra: Spectral Theor...  

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: https://www.seh.ox.ac.uk/people/tom-c...

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