Oxford Linear Algebra: Rank Nullity Theorem

University of Oxford mathematician Dr Tom Crawford introduces the concepts of rank and nullity for a linear transformation, before going through a full step-by-step proof of the Rank Nullity Theorem. Check out ProPrep with a 30-day free trial: www.proprep.uk/info/TOM-Crawford
Links to the other videos mentioned:
Linear Transformations: • Oxford Linear Algebra:...
Dimension Formula: • Oxford Linear Algebra:...
Spanning, Basis & Linear Independence: • Oxford Linear Algebra:...
Subspace Test: • Oxford Linear Algebra:...
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: api.proprep.com/course/downlo...
You can also find several video lectures from ProPrep explaining the content covered in the video here: www.proprep.com/courses/all/l...
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
The video begins with the definition of the kernel and image of a linear transformation, as well as their dimensions as the nullity and rank of the linear map. Next is a fully worked example of a map in R^2 with an explicit calculation of the kernel, image, nullity and rank.
In the second part of the video the Rank Nullity Theorem is presented: the dimension of the starting vector space of a linear map is equal to the rank of the linear map (dimension of its image) plus the nullity of the map (dimension of its kernel). A full step-by-step proof is then presented by constructing a basis for each component.
Watch the 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:...
Vector Space Axioms: • Oxford Linear Algebra:...
Subspace Test: • Oxford Linear Algebra:...
Basis, Spanning and Linear Independence: • Oxford Linear Algebra:...
Dimension Formula: • Oxford Linear Algebra:...
Direct Sum: • Oxford Linear Algebra:...
Linear Transformations: • Oxford Linear Algebra:...
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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Get your Tom Rocks Maths merchandise here:
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Check out Proprep with a 30-day free trial here: www.proprep.uk/info/TOM-Crawford

[@TomRocksMaths]

Remember to check out ProPrep with a 30-day free trial here: www.proprep.uk/info/TOM-Crawford

[@KingOfGamesss]

Hi Dr Tom found a little bit of random maths for you in this cartoon series 'Teenage Mutant Ninja Turtles Season 2 Episode 21 - April's Artifact' (SKIP TO 13:45 to 14:15) ua-cam.com/video/Gs9E8i2-fqs/v-deo.html

[@sebastian20315]

he doesn't look like a guy who would do maths, but he is a really nice and smart guy

[@yanbopan]

The best rank nullity theorem video I have ever seen ;)

[@mayankmudgal1924]

You are a great teacher 😊

[@primefactor5417]

Sir .. please make a lecture on how to extend a basis to another basis ...

[@peregrinef3203]

Did you test 10 to see if 11 was, indeed, the minimum number needed?

[@patrickkendall5282]

Hey dr tom

[@rekhachhaya7158]

You teach in oxford??

[@user-sz5ce1qs5r]

OMG, that amazing

[@Ghost____Rider]

What is?

[@ProGamer-ru2gd]

you should make a video about jee advance soon

[@baltzarbrat]

Sir can you solve
Jee advanced maths question paper ?

[@Hydrogn_]

I thought I could learn something but I'm too young to understand this :Σ

[@msferferi-mf9ki]

دلم خواست

[@MariadeLourdesAniesSanch-ze7hf]

I think finish
but my color is pink
because is He

[@IamBraads]

Bad hair day again 🫤

[@alphaa]

I'm on that side of UA-cam again